Documentation Verification Report

Function

πŸ“ Source: Mathlib/Data/Set/Function.lean

Statistics

MetricCount
DefinitionsinvFunOn
1
Theoremsimage, bijOn, bijOn', bijOn_image, bijOn_swap, bijOn_symm, bijOn_symm_image, image_eq_iff_bijOn, invOn, bijOn_univ, comp_injOn, injOn, injOn_range, leftInvOn, rightInvOn_range, rightInvOn, bijOn_image, bijOn_range, injOn_image, injOn_preimage, injOn_range, mapsTo_image, mapsTo_image_right, mapsTo_preimage, mapsTo_range, surjOn_image, surjOn_range, surjOn, apply_eq_of_range_eq_singleton, insert_injOn, invFunOn_apply_eq, invFunOn_apply_mem, invFunOn_eq, invFunOn_image_image_subset, invFunOn_injOn_image, invFunOn_mem, invFunOn_neg, invFunOn_pos, leftInvOn_invFunOn_of_subset_image_image, update_comp_eq_of_notMem_range, update_comp_eq_of_notMem_range', bijective, comp, compl, equiv_symm, exists, exists_extend, exists_extend_of_subset, extendDomain, forall, image_eq, injOn, insert, insert_iff, inter, inter_mapsTo, invOn_invFunOn, iterate, mapsTo, mk, of_equiv_symm, prodMap, sdiff_singleton, subset_left, subset_range, subset_right, surjOn, union, bijOn_iff, cancel_left, cancel_right, comp_eq, comp_left, comp_leftβ‚‚, comp_right, eq_of_mem, image_eq, image_eq_self, injOn_iff, inter_preimage_eq, mapsTo_iff, mono, surjOn_iff, trans, union, bijOn_image, cancel_left, comp, comp_iff, eq_iff, exists_subset_injOn_subset_range_eq, image, imageFactorization_injective, image_diff, image_diff_subset, image_eq_image_iff, image_inter, image_of_comp, image_ssubset_image_iff, image_subset_image_iff, injective_iff, invFunOn_image, iterate, leftInvOn_invFunOn, mem_image_iff, mem_of_mem_image, mono, ne, ne_iff, of_comp, preimage_image_inter, prodMap, rightInvOn_of_leftInvOn, bijOn, comp, extendDomain, mono, prodMap, comp, congr_left, congr_right, eq, eqOn, extendDomain, image_image, image_image', image_inter, image_inter', injOn, mapsTo, mono, prodMap, rightInvOn_image, surjOn, comp, comp_left, comp_right, extendDomain, image_subset, insert, inter, inter_bijOn, inter_inter, iterate, iterate_restrict, mem_iff, mono, mono_left, mono_right, nonempty, prodMap, subset_preimage, surjOn_compl, union, union_union, comp, congr_left, congr_right, eq, eqOn, extendDomain, mapsTo, mono, prodMap, surjOn, injOn, bijOn_subset, cancel_right, comap_nonempty, comp, comp_left, comp_right, exists_bijOn_subset, exists_subset_injOn_image_eq, extendDomain, forall, image_eq_of_mapsTo, image_invFunOn_image, image_invFunOn_image_of_subset, image_preimage, inter, inter_inter, invOn_invFunOn, iterate, leftInvOn_of_rightInvOn, mapsTo_compl, mapsTo_invFunOn, mono, of_comp, prodMap, rightInvOn_invFunOn, subset_range, surjective, union, union_union, bijOn_comm, bijOn_comp_iff, bijOn_empty, bijOn_empty_iff_left, bijOn_empty_iff_right, bijOn_id, bijOn_image_image, bijOn_of_subsingleton, bijOn_of_subsingleton', bijOn_singleton, bijOn_univ, bijective_iff_bijOn_univ, eqOn_comm, eqOn_comp_right_iff, eqOn_empty, eqOn_of_leftInvOn_of_rightInvOn, eqOn_range, eqOn_refl, eqOn_singleton, eqOn_union, eqOn_univ, exists_eq_graphOn, exists_eq_graphOn_image_fst, exists_image_eq_and_injOn, exists_image_eq_injOn_of_subset_range, exists_injOn_iff_injective, exists_subset_bijOn, graphOn_univ_inj, graphOn_univ_injective, imageFactorization_injective_iff, image_diff_of_injOn, image_eq_iff_surjOn_mapsTo, image_eq_preimage_of_leftInvOn_injOn, image_eq_preimage_of_leftInvOn_injOn_mapsTo, injOn_empty, injOn_id, injOn_iff_invFunOn_image_image_eq_self, injOn_insert, injOn_of_eq_iff_eq, injOn_of_injective, injOn_of_subsingleton, injOn_pair, injOn_preimage, injOn_singleton, injOn_subtype_val, injOn_union, injOn_univ, injective_iff_injOn_univ, invOn_id, leftInvOn_id, mapsTo_empty, mapsTo_empty_iff, mapsTo_id, mapsTo_iff_image_subset, mapsTo_iff_subset_preimage, mapsTo_image_iff, mapsTo_inter, mapsTo_of_subsingleton, mapsTo_of_subsingleton', mapsTo_prodMap_diagonal, mapsTo_range, mapsTo_range_iff, mapsTo_singleton, mapsTo_union, mapsTo_univ, mapsTo_univ_iff, mapsTo_univ_iff_range_subset, preimage_invFun_of_mem, preimage_invFun_of_notMem, rightInvOn_id, surjOn_comp_iff, surjOn_empty, surjOn_empty_iff, surjOn_id, surjOn_iff_exists_bijOn_subset, surjOn_iff_exists_map_subtype, surjOn_image, surjOn_of_subsingleton, surjOn_of_subsingleton', surjOn_singleton, surjOn_univ, surjOn_univ_of_subsingleton_nonempty, surjective_iff_surjOn_univ, coind_bijective, coind_surjective
280
Total281

Disjoint

Theorems

NameKindAssumesProvesValidatesDepends On
image πŸ“–mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
Set.InjOn
Set.instHasSubset
Disjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
Set.image
β€”Set.disjoint_iff_inter_eq_empty
Set.InjOn.image_inter
Set.image_empty

Equiv

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn πŸ“–mathematicalSet
Set.instMembership
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
β€”bijOn'
apply_symm_apply
bijOn' πŸ“–mathematicalSet.MapsTo
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
β€”Function.Injective.injOn
injective
apply_symm_apply
bijOn_image πŸ“–mathematicalβ€”Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
Set.image
β€”Set.InjOn.bijOn_image
Function.Injective.injOn
injective
bijOn_swap πŸ“–mathematicalSet
Set.instMembership
Set.BijOn
DFunLike.coe
Perm
EquivLike.toFunLike
instEquivLike
swap
β€”bijOn
eq_or_ne
swap_self
swap_apply_left
swap_apply_right
swap_apply_of_ne_of_ne
bijOn_symm πŸ“–mathematicalβ€”Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
β€”Set.bijOn_comm
invOn
bijOn_symm_image πŸ“–mathematicalβ€”Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
Set.image
β€”Set.BijOn.symm
invOn
bijOn_image
image_eq_iff_bijOn πŸ“–mathematicalβ€”Set.image
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
Set.BijOn
β€”Set.MapsTo.mono_right
Set.mapsTo_image
Eq.subset
Set.instReflSubset
Function.Injective.injOn
injective
Set.surjOn_image
Set.BijOn.image_eq
invOn πŸ“–mathematicalβ€”Set.InvOn
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
β€”Function.LeftInverse.leftInvOn
rightInverse_symm
leftInverse_symm

Function

Definitions

NameCategoryTheorems
invFunOn πŸ“–CompOp
20 mathmath: Set.SurjOn.invOn_invFunOn, Set.SurjOn.mapsTo_invFunOn, invFunOn_mem, Set.SurjOn.image_invFunOn_image_of_subset, Set.InjOn.invFunOn_image, invFunOn_injOn_image, Set.SurjOn.image_invFunOn_image, Set.SurjOn.rightInvOn_invFunOn, invFunOn_apply_mem, invFunOn_eq, invFunOn_pos, invFunOn_neg, invFunOn_apply_eq, leftInvOn_invFunOn_of_subset_image_image, Set.InjOn.leftInvOn_invFunOn, Set.BijOn.invOn_invFunOn, Set.BijOn.toPartialEquiv_symm_apply, invFunOn_image_image_subset, Set.injOn_iff_invFunOn_image_image_eq_self, Set.SurjOn.bijOn_subset

Theorems

NameKindAssumesProvesValidatesDepends On
apply_eq_of_range_eq_singleton πŸ“–β€”Set.range
Set
Set.instSingletonSet
β€”β€”Set.mem_range_self
insert_injOn πŸ“–mathematicalβ€”Set.InjOn
Set
Set.instInsert
Compl.compl
Set.instCompl
β€”β€”
invFunOn_apply_eq πŸ“–mathematicalSet
Set.instMembership
invFunOnβ€”invFunOn_eq
invFunOn_apply_mem πŸ“–mathematicalSet
Set.instMembership
Set
Set.instMembership
invFunOn
β€”invFunOn_mem
invFunOn_eq πŸ“–mathematicalSet
Set.instMembership
invFunOnβ€”invFunOn_pos
invFunOn_image_image_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.image
invFunOn
β€”invFunOn_apply_mem
invFunOn_injOn_image πŸ“–mathematicalβ€”Set.InjOn
invFunOn
Set.image
β€”invFunOn_apply_eq
invFunOn_mem πŸ“–mathematicalSet
Set.instMembership
Set
Set.instMembership
invFunOn
β€”invFunOn_pos
invFunOn_neg πŸ“–mathematicalSet
Set.instMembership
invFunOnβ€”invFunOn.eq_1
invFunOn_pos πŸ“–mathematicalSet
Set.instMembership
Set
Set.instMembership
invFunOn
β€”invFunOn.eq_1
leftInvOn_invFunOn_of_subset_image_image πŸ“–mathematicalSet
Set.instHasSubset
Set.image
invFunOn
Set.LeftInvOn
invFunOn
β€”invFunOn_apply_eq
update_comp_eq_of_notMem_range πŸ“–mathematicalSet
Set.instMembership
Set.range
updateβ€”update_comp_eq_of_notMem_range'
update_comp_eq_of_notMem_range' πŸ“–mathematicalSet
Set.instMembership
Set.range
updateβ€”update_comp_eq_of_forall_ne'

Function.Bijective

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn_univ πŸ“–mathematicalFunction.BijectiveSet.BijOn
Set.univ
β€”Set.bijOn_univ

Function.Injective

Theorems

NameKindAssumesProvesValidatesDepends On
comp_injOn πŸ“–mathematicalSet.InjOnSet.InjOnβ€”Set.InjOn.comp
injOn
Set.mapsTo_univ
injOn πŸ“–mathematicalβ€”Set.InjOnβ€”Set.injOn_of_injective
injOn_range πŸ“–mathematicalβ€”Set.InjOn
Set.range
β€”β€”

Function.LeftInverse

Theorems

NameKindAssumesProvesValidatesDepends On
leftInvOn πŸ“–mathematicalβ€”Set.LeftInvOnβ€”β€”
rightInvOn_range πŸ“–mathematicalβ€”Set.RightInvOn
Set.range
β€”Set.forall_mem_range

Function.RightInverse

Theorems

NameKindAssumesProvesValidatesDepends On
rightInvOn πŸ“–mathematicalβ€”Set.RightInvOnβ€”β€”

Function.Semiconj

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn_image πŸ“–mathematicalFunction.Semiconj
Set.BijOn
Set.InjOn
Set.BijOn
Set.image
β€”mapsTo_image
Set.BijOn.mapsTo
injOn_image
Set.BijOn.injOn
Set.BijOn.image_eq
surjOn_image
Set.BijOn.surjOn
bijOn_range πŸ“–mathematicalFunction.Semiconj
Function.Bijective
Set.BijOn
Set.range
β€”Set.image_univ
bijOn_image
Function.Bijective.bijOn_univ
Function.Injective.injOn
injOn_image πŸ“–mathematicalFunction.Semiconj
Set.InjOn
Set.image
Set.InjOn
Set.image
β€”Set.mem_image_of_mem
eq
injOn_preimage πŸ“–mathematicalFunction.Semiconj
Set.InjOn
Set.preimage
Set.InjOn
Set.preimage
β€”eq
injOn_range πŸ“–mathematicalFunction.Semiconj
Set.InjOn
Set.range
Set.InjOn
Set.range
β€”Set.image_univ
injOn_image
Function.Injective.injOn
mapsTo_image πŸ“–mathematicalFunction.Semiconj
Set.MapsTo
Set.MapsTo
Set.image
β€”β€”
mapsTo_image_right πŸ“–mathematicalFunction.Semiconj
Set.MapsTo
Set.MapsTo
Set.image
β€”Set.mapsTo_image_iff
mapsTo_preimage πŸ“–mathematicalFunction.Semiconj
Set.MapsTo
Set.MapsTo
Set.preimage
β€”β€”
mapsTo_range πŸ“–mathematicalFunction.SemiconjSet.MapsTo
Set.range
β€”β€”
surjOn_image πŸ“–mathematicalFunction.Semiconj
Set.SurjOn
Set.SurjOn
Set.image
β€”Set.mem_image_of_mem
surjOn_range πŸ“–mathematicalFunction.SemiconjSet.SurjOn
Set.range
β€”Set.image_univ
surjOn_image
Function.Surjective.surjOn

Function.Surjective

Theorems

NameKindAssumesProvesValidatesDepends On
surjOn πŸ“–mathematicalβ€”Set.SurjOn
Set.univ
β€”Set.SurjOn.mono
Set.Subset.rfl
Set.subset_univ
Set.surjOn_univ

Set

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn_comm πŸ“–mathematicalInvOnBijOnβ€”BijOn.symm
InvOn.symm
bijOn_comp_iff πŸ“–mathematicalInjOnBijOn
image
β€”β€”
bijOn_empty πŸ“–mathematicalβ€”BijOn
Set
instEmptyCollection
β€”mapsTo_empty
injOn_empty
surjOn_empty
bijOn_empty_iff_left πŸ“–mathematicalβ€”BijOn
Set
instEmptyCollection
β€”BijOn.mapsTo
bijOn_empty
bijOn_empty_iff_right πŸ“–mathematicalβ€”BijOn
Set
instEmptyCollection
β€”BijOn.surjOn
bijOn_empty
bijOn_id πŸ“–mathematicalβ€”BijOnβ€”mapsTo_id
injOn_id
surjOn_id
bijOn_image_image πŸ“–mathematicalBijOn
InjOn
image
BijOn
image
β€”β€”
bijOn_of_subsingleton πŸ“–mathematicalβ€”BijOnβ€”bijOn_of_subsingleton'
bijOn_of_subsingleton' πŸ“–mathematicalNonemptyBijOnβ€”mapsTo_of_subsingleton'
injOn_of_subsingleton
surjOn_of_subsingleton'
bijOn_singleton πŸ“–mathematicalβ€”BijOn
Set
instSingletonSet
β€”image_singleton
bijOn_univ πŸ“–mathematicalβ€”BijOn
univ
Function.Bijective
β€”β€”
bijective_iff_bijOn_univ πŸ“–mathematicalβ€”Function.Bijective
BijOn
univ
β€”bijOn_univ
eqOn_comm πŸ“–mathematicalβ€”EqOnβ€”EqOn.symm
eqOn_comp_right_iff πŸ“–mathematicalβ€”EqOn
image
β€”SurjOn.cancel_right
surjOn_image
mapsTo_image
eqOn_empty πŸ“–mathematicalβ€”EqOn
Set
instEmptyCollection
β€”β€”
eqOn_of_leftInvOn_of_rightInvOn πŸ“–mathematicalLeftInvOn
RightInvOn
MapsTo
EqOnβ€”β€”
eqOn_range πŸ“–mathematicalβ€”EqOn
range
β€”forall_mem_range
eqOn_refl πŸ“–mathematicalβ€”EqOnβ€”β€”
eqOn_singleton πŸ“–mathematicalβ€”EqOn
Set
instSingletonSet
β€”β€”
eqOn_union πŸ“–mathematicalβ€”EqOn
Set
instUnion
β€”forallβ‚‚_or_left
eqOn_univ πŸ“–mathematicalβ€”EqOn
univ
β€”β€”
exists_eq_graphOn πŸ“–mathematicalβ€”Set
graphOn
InjOn
β€”image_fst_graphOn
exists_eq_graphOn_image_fst
exists_eq_graphOn_image_fst πŸ“–mathematicalβ€”graphOn
image
InjOn
β€”InjOn.image_of_comp
injOn_id
forall_mem_image
graphOn.eq_1
image_image
EqOn.image_eq_self
Function.sometimes_spec
exists_image_eq_and_injOn πŸ“–mathematicalβ€”Set
image
InjOn
β€”exists_subset_bijOn
BijOn.image_eq
BijOn.injOn
exists_image_eq_injOn_of_subset_range πŸ“–mathematicalSet
instHasSubset
range
Set
image
InjOn
β€”exists_image_eq_and_injOn
image_preimage_eq_of_subset
exists_injOn_iff_injective πŸ“–mathematicalβ€”InjOn
Elem
β€”InjOn.injective
CanLift.prf
PiSetCoe.canLift'
injOn_iff_injective
exists_subset_bijOn πŸ“–mathematicalβ€”Set
instHasSubset
BijOn
image
β€”surjOn_iff_exists_bijOn_subset
surjOn_image
graphOn_univ_inj πŸ“–mathematicalβ€”graphOn
univ
β€”β€”
graphOn_univ_injective πŸ“–mathematicalβ€”Set
graphOn
univ
β€”β€”
imageFactorization_injective_iff πŸ“–mathematicalβ€”Elem
image
imageFactorization
InjOn
β€”InjOn.imageFactorization_injective
image_diff_of_injOn πŸ“–mathematicalInjOn
Set
instHasSubset
image
Set
instSDiff
β€”InjOn.image_diff_subset
image_eq_iff_surjOn_mapsTo πŸ“–mathematicalβ€”image
SurjOn
MapsTo
β€”surjOn_image
mapsTo_image
SurjOn.image_eq_of_mapsTo
image_eq_preimage_of_leftInvOn_injOn πŸ“–mathematicalLeftInvOn
InjOn
preimage
image
preimage
β€”ext
mem_preimage
InjOn.rightInvOn_of_leftInvOn
mapsTo_preimage
image_eq_preimage_of_leftInvOn_injOn_mapsTo πŸ“–mathematicalLeftInvOn
InjOn
preimage
image
preimage
β€”image_eq_preimage_of_leftInvOn_injOn
injOn_empty πŸ“–mathematicalβ€”InjOn
Set
instEmptyCollection
β€”Subsingleton.injOn
subsingleton_empty
injOn_id πŸ“–mathematicalβ€”InjOnβ€”Function.Injective.injOn
injOn_iff_invFunOn_image_image_eq_self πŸ“–mathematicalβ€”InjOn
image
Function.invFunOn
β€”InjOn.invFunOn_image
Subset.rfl
LeftInvOn.injOn
Function.leftInvOn_invFunOn_of_subset_image_image
Eq.subset
instReflSubset
injOn_insert πŸ“–mathematicalSet
instMembership
InjOn
Set
instInsert
instMembership
image
β€”union_singleton
injOn_union
disjoint_singleton_right
injOn_of_eq_iff_eq πŸ“–mathematicalβ€”InjOnβ€”β€”
injOn_of_injective πŸ“–mathematicalβ€”InjOnβ€”β€”
injOn_of_subsingleton πŸ“–mathematicalβ€”InjOnβ€”Function.Injective.injOn
Function.injective_of_subsingleton
injOn_pair πŸ“–mathematicalβ€”InjOn
Set
instInsert
instSingletonSet
β€”β€”
injOn_preimage πŸ“–mathematicalSet
instHasSubset
powerset
range
InjOn
Set
preimage
β€”preimage_eq_preimage'
injOn_singleton πŸ“–mathematicalβ€”InjOn
Set
instSingletonSet
β€”Subsingleton.injOn
subsingleton_singleton
injOn_subtype_val πŸ“–mathematicalβ€”InjOnβ€”Function.Injective.injOn
Subtype.coe_injective
injOn_union πŸ“–mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
InjOn
Set
instUnion
β€”InjOn.mono
subset_union_left
subset_union_right
Disjoint.le_bot
injOn_univ πŸ“–mathematicalβ€”InjOn
univ
β€”β€”
injective_iff_injOn_univ πŸ“–mathematicalβ€”InjOn
univ
β€”injOn_univ
invOn_id πŸ“–mathematicalβ€”InvOnβ€”leftInvOn_id
rightInvOn_id
leftInvOn_id πŸ“–mathematicalβ€”LeftInvOnβ€”β€”
mapsTo_empty πŸ“–mathematicalβ€”MapsTo
Set
instEmptyCollection
β€”β€”
mapsTo_empty_iff πŸ“–mathematicalβ€”MapsTo
Set
instEmptyCollection
β€”β€”
mapsTo_id πŸ“–mathematicalβ€”MapsToβ€”β€”
mapsTo_iff_image_subset πŸ“–mathematicalβ€”MapsTo
Set
instHasSubset
image
β€”image_subset_iff
mapsTo_iff_subset_preimage πŸ“–mathematicalβ€”MapsTo
Set
instHasSubset
preimage
β€”β€”
mapsTo_image_iff πŸ“–mathematicalβ€”MapsTo
image
β€”β€”
mapsTo_inter πŸ“–mathematicalβ€”MapsTo
Set
instInter
β€”MapsTo.mono
Subset.refl
inter_subset_left
inter_subset_right
MapsTo.inter
mapsTo_of_subsingleton πŸ“–mathematicalβ€”MapsToβ€”mapsTo_of_subsingleton'
mapsTo_of_subsingleton' πŸ“–mathematicalNonemptyMapsToβ€”Subsingleton.mem_iff_nonempty
mapsTo_prodMap_diagonal πŸ“–mathematicalβ€”MapsTo
diagonal
β€”mapsTo_iff_subset_preimage
diagonal_subset_iff
mapsTo_range πŸ“–mathematicalβ€”MapsTo
range
β€”MapsTo.mono
mapsTo_image
Subset.refl
image_subset_range
mapsTo_range_iff πŸ“–mathematicalβ€”MapsTo
range
Set
instMembership
β€”forall_mem_range
mapsTo_singleton πŸ“–mathematicalβ€”MapsTo
Set
instSingletonSet
instMembership
β€”mapsTo_iff_subset_preimage
singleton_subset_iff
mapsTo_union πŸ“–mathematicalβ€”MapsTo
Set
instUnion
β€”MapsTo.mono
subset_union_left
Subset.refl
subset_union_right
MapsTo.union
mapsTo_univ πŸ“–mathematicalβ€”MapsTo
univ
β€”β€”
mapsTo_univ_iff πŸ“–mathematicalβ€”MapsTo
univ
Set
instMembership
β€”mem_univ
mapsTo_univ_iff_range_subset πŸ“–mathematicalβ€”MapsTo
univ
Set
instHasSubset
range
β€”mapsTo_univ_iff
range_subset_iff
preimage_invFun_of_mem πŸ“–mathematicalSet
instMembership
preimage
Function.invFun
Set
instUnion
image
Compl.compl
instCompl
range
β€”ext
em
Function.leftInverse_invFun
Function.Injective.mem_set_image
Function.invFun_neg
preimage_invFun_of_notMem πŸ“–mathematicalSet
instMembership
preimage
Function.invFun
image
β€”ext
em
mem_preimage
Function.leftInverse_invFun
Function.Injective.mem_set_image
image_subset_range
Function.invFun_neg
rightInvOn_id πŸ“–mathematicalβ€”RightInvOnβ€”β€”
surjOn_comp_iff πŸ“–mathematicalβ€”SurjOn
image
β€”SurjOn.of_comp
mapsTo_image
SurjOn.comp
surjOn_image
surjOn_empty πŸ“–mathematicalβ€”SurjOn
Set
instEmptyCollection
β€”empty_subset
surjOn_empty_iff πŸ“–mathematicalβ€”SurjOn
Set
instEmptyCollection
β€”image_empty
surjOn_id πŸ“–mathematicalβ€”SurjOnβ€”image_congr
image_id'
instReflSubset
surjOn_iff_exists_bijOn_subset πŸ“–mathematicalβ€”SurjOn
Set
instHasSubset
BijOn
β€”eq_empty_or_nonempty
empty_subset
bijOn_empty
SurjOn.comap_nonempty
MapsTo.image_subset
SurjOn.mapsTo_invFunOn
SurjOn.bijOn_subset
SurjOn.mono
Subset.refl
BijOn.surjOn
surjOn_iff_exists_map_subtype πŸ“–mathematicalβ€”SurjOn
Set
instHasSubset
Elem
instMembership
β€”mapsTo_image
surjective_mapsTo_image_restrict
surjOn_image πŸ“–mathematicalβ€”SurjOn
image
β€”Subset.rfl
surjOn_of_subsingleton πŸ“–mathematicalβ€”SurjOnβ€”surjOn_of_subsingleton'
surjOn_of_subsingleton' πŸ“–mathematicalNonemptySurjOnβ€”Subsingleton.mem_iff_nonempty
Nonempty.image
surjOn_singleton πŸ“–mathematicalβ€”SurjOn
Set
instSingletonSet
instMembership
image
β€”singleton_subset_iff
surjOn_univ πŸ“–mathematicalβ€”SurjOn
univ
β€”image_univ
surjOn_univ_of_subsingleton_nonempty πŸ“–mathematicalβ€”SurjOn
univ
Nonempty
β€”nonempty_unique
univ_unique
image_congr
surjective_iff_surjOn_univ πŸ“–mathematicalβ€”SurjOn
univ
β€”surjOn_univ

Set.BijOn

Theorems

NameKindAssumesProvesValidatesDepends On
bijective πŸ“–mathematicalSet.BijOnFunction.Bijective
Set.Elem
Set.MapsTo.restrict
mapsTo
β€”mapsTo
injOn
surjOn
comp πŸ“–mathematicalSet.BijOnSet.BijOnβ€”Set.MapsTo.comp
mapsTo
Set.InjOn.comp
injOn
Set.SurjOn.comp
surjOn
compl πŸ“–mathematicalSet.BijOn
Function.Bijective
Set.BijOn
Compl.compl
Set
Set.instCompl
β€”Set.SurjOn.mapsTo_compl
surjOn
Function.Injective.injOn
Set.MapsTo.surjOn_compl
mapsTo
equiv_symm πŸ“–mathematicalSet.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”Equiv.bijOn_symm
exists πŸ“–mathematicalSet.BijOnSet
Set.instMembership
β€”surjOn
mapsTo
exists_extend πŸ“–mathematicalSet.BijOn
Set
Set.instHasSubset
Set.range
Set
Set.instHasSubset
Set.BijOn
β€”exists_extend_of_subset
Set.subset_univ
Set.image_univ
exists_extend_of_subset πŸ“–mathematicalSet.BijOn
Set
Set.instHasSubset
Set.SurjOn
Set
Set.instHasSubset
Set.BijOn
β€”Set.exists_subset_bijOn
Set.subset_union_left
Set.union_subset
HasSubset.Subset.trans
Set.instIsTransSubset
Set.diff_subset
Set.inter_subset_left
Set.mapsTo_iff_image_subset
Set.image_union
image_eq
Set.image_inter_preimage
Set.image_diff_preimage
Set.union_subset_iff
Set.inter_subset_right
Set.injOn_union
Disjoint.mono_left
mapsTo
Disjoint.symm
Set.subset_diff
injOn
Set.union_diff_self
extendDomain πŸ“–mathematicalSet.BijOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Set.BijOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Set.MapsTo.extendDomain
mapsTo
Function.Injective.injOn
Equiv.injective
Set.SurjOn.extendDomain
surjOn
forall πŸ“–β€”Set.BijOnβ€”β€”mapsTo
surjOn
image_eq πŸ“–mathematicalSet.BijOnSet.imageβ€”Set.SurjOn.image_eq_of_mapsTo
surjOn
mapsTo
injOn πŸ“–mathematicalSet.BijOnSet.InjOnβ€”β€”
insert πŸ“–mathematicalSet.BijOn
Set
Set.instMembership
Set.BijOn
Set
Set.instInsert
β€”insert_iff
mapsTo
insert_iff πŸ“–mathematicalSet
Set.instMembership
Set.BijOn
Set
Set.instInsert
β€”image_eq
Set.image_insert_eq
Set.diff_singleton_eq_self
Set.InjOn.eq_iff
injOn
Set.insert_diff_of_mem
Set.instReflSubset
Set.InjOn.mono
Set.subset_insert
Set.insert_eq
union
Set.bijOn_singleton
Set.injOn_insert
mapsTo
inter πŸ“–mathematicalSet.BijOn
Set.InjOn
Set
Set.instUnion
Set.BijOn
Set
Set.instInter
β€”Set.MapsTo.inter_inter
mapsTo
Set.InjOn.mono
Set.inter_subset_left
injOn
Set.SurjOn.inter_inter
surjOn
inter_mapsTo πŸ“–mathematicalSet.BijOn
Set.MapsTo
Set
Set.instHasSubset
Set.instInter
Set.preimage
Set.BijOn
Set
Set.instInter
β€”Set.MapsTo.inter_inter
mapsTo
Set.InjOn.mono
Set.inter_subset_left
injOn
surjOn
invOn_invFunOn πŸ“–mathematicalSet.BijOnSet.InvOn
Function.invFunOn
β€”Set.InjOn.leftInvOn_invFunOn
injOn
Set.SurjOn.rightInvOn_invFunOn
surjOn
iterate πŸ“–mathematicalSet.BijOnSet.BijOn
Nat.iterate
β€”Set.bijOn_id
comp
mapsTo πŸ“–mathematicalSet.BijOnSet.MapsToβ€”β€”
mk πŸ“–mathematicalSet.MapsTo
Set.InjOn
Set.SurjOn
Set.BijOnβ€”β€”
of_equiv_symm πŸ“–mathematicalSet.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
β€”Equiv.bijOn_symm
prodMap πŸ“–mathematicalSet.BijOnSet.BijOn
SProd.sprod
Set
Set.instSProd
β€”Set.MapsTo.prodMap
mapsTo
Set.InjOn.prodMap
injOn
Set.SurjOn.prodMap
surjOn
sdiff_singleton πŸ“–mathematicalSet.BijOn
Set
Set.instMembership
Set.BijOn
Set
Set.instSDiff
Set.instSingletonSet
β€”Set.InjOn.image_diff
injOn
image_eq
Set.inter_eq_self_of_subset_right
Set.image_singleton
subset_left
Set.diff_subset
subset_left πŸ“–mathematicalSet.BijOn
Set
Set.instHasSubset
Set.BijOn
Set.image
β€”Set.InjOn.bijOn_image
Set.InjOn.mono
injOn
subset_range πŸ“–mathematicalSet.BijOnSet
Set.instHasSubset
Set.range
β€”Set.SurjOn.subset_range
surjOn
subset_right πŸ“–mathematicalSet.BijOn
Set
Set.instHasSubset
Set.BijOn
Set
Set.instInter
Set.preimage
β€”Set.inter_subset_right
Set.InjOn.mono
Set.inter_subset_left
injOn
surjOn
surjOn πŸ“–mathematicalSet.BijOnSet.SurjOnβ€”β€”
union πŸ“–mathematicalSet.BijOn
Set.InjOn
Set
Set.instUnion
Set.BijOn
Set
Set.instUnion
β€”Set.MapsTo.union_union
mapsTo
Set.SurjOn.union_union
surjOn

Set.EqOn

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn_iff πŸ“–mathematicalSet.EqOnSet.BijOnβ€”Set.BijOn.congr
symm
cancel_left πŸ“–mathematicalSet.EqOn
Set.InjOn
Set.MapsTo
Set.EqOnβ€”β€”
cancel_right πŸ“–mathematicalSet.EqOn
Set.SurjOn
Set.EqOnβ€”β€”
comp_eq πŸ“–β€”Set.EqOn
Set.range
β€”β€”Set.eqOn_range
comp_left πŸ“–mathematicalSet.EqOnSet.EqOnβ€”β€”
comp_leftβ‚‚ πŸ“–mathematicalSet.EqOnSet.EqOnβ€”β€”
comp_right πŸ“–mathematicalSet.EqOn
Set.MapsTo
Set.EqOnβ€”β€”
eq_of_mem πŸ“–β€”Set.EqOn
Set
Set.instMembership
β€”β€”β€”
image_eq πŸ“–mathematicalSet.EqOnSet.imageβ€”β€”
image_eq_self πŸ“–mathematicalSet.EqOnSet.imageβ€”β€”
injOn_iff πŸ“–mathematicalSet.EqOnSet.InjOnβ€”Set.InjOn.congr
symm
inter_preimage_eq πŸ“–mathematicalSet.EqOnSet
Set.instInter
Set.preimage
β€”β€”
mapsTo_iff πŸ“–mathematicalSet.EqOnSet.MapsToβ€”Set.MapsTo.congr
symm
mono πŸ“–mathematicalSet
Set.instHasSubset
Set.EqOn
Set.EqOnβ€”β€”
surjOn_iff πŸ“–mathematicalSet.EqOnSet.SurjOnβ€”Set.SurjOn.congr
symm
trans πŸ“–mathematicalSet.EqOnSet.EqOnβ€”β€”
union πŸ“–mathematicalSet.EqOnSet.EqOn
Set
Set.instUnion
β€”Set.eqOn_union

Set.InjOn

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn_image πŸ“–mathematicalSet.InjOnSet.BijOn
Set.image
β€”Set.mapsTo_image
Set.Subset.refl
cancel_left πŸ“–mathematicalSet.InjOn
Set.MapsTo
Set.EqOnβ€”Set.EqOn.cancel_left
Set.EqOn.comp_left
comp πŸ“–mathematicalSet.InjOn
Set.MapsTo
Set.InjOnβ€”β€”
comp_iff πŸ“–mathematicalSet.InjOnSet.InjOn
Set.image
β€”image_of_comp
comp
Set.mapsTo_image
eq_iff πŸ“–β€”Set.InjOn
Set
Set.instMembership
β€”β€”β€”
exists_subset_injOn_subset_range_eq πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set
Set.instHasSubset
Set.image
Set.InjOn
β€”Set.BijOn.exists_extend_of_subset
bijOn_image
Set.image_mono
Set.Subset.rfl
Set.BijOn.image_eq
Set.BijOn.injOn
image πŸ“–mathematicalSet.InjOnSet.InjOn
Set
Set.image
Set.powerset
β€”preimage_image_inter
imageFactorization_injective πŸ“–mathematicalSet.InjOnSet.Elem
Set.image
Set.imageFactorization
β€”eq_iff
image_diff πŸ“–mathematicalSet.InjOnSet.image
Set
Set.instSDiff
Set.instInter
β€”subset_antisymm
Set.instAntisymmSubset
Set.subset_diff
Set.image_mono
Set.diff_subset
Disjoint.image
Set.disjoint_sdiff_inter
Set.inter_subset_left
Set.diff_subset_iff
Set.image_union
Set.inter_union_diff
subset_refl
Set.instReflSubset
image_diff_subset πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.image
Set
Set.instSDiff
β€”image_diff
Set.inter_eq_self_of_subset_right
image_eq_image_iff πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.imageβ€”eq_iff
image
image_inter πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.image
Set
Set.instInter
β€”Set.Subset.antisymm
Set.image_inter_subset
image_of_comp πŸ“–mathematicalSet.InjOnSet.InjOn
Set.image
β€”Set.forall_mem_image
image_ssubset_image_iff πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set
Set.instHasSSubset
Set.image
β€”image_subset_image_iff
image_subset_image_iff πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set
Set.instHasSubset
Set.image
β€”preimage_image_inter
Set.inter_subset_inter_left
Set.preimage_mono
Set.image_mono
injective_iff πŸ“–β€”Set.InjOn
Set
Set.instHasSubset
Set.range
β€”β€”Function.Injective.of_comp
invFunOn_image πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.image
Function.invFunOn
β€”Set.LeftInvOn.image_image'
leftInvOn_invFunOn
iterate πŸ“–mathematicalSet.InjOn
Set.MapsTo
Set.InjOn
Nat.iterate
β€”Set.injOn_id
comp
leftInvOn_invFunOn πŸ“–mathematicalSet.InjOnSet.LeftInvOn
Function.invFunOn
β€”Function.invFunOn_apply_mem
Function.invFunOn_apply_eq
mem_image_iff πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.instMembership
Set
Set.instMembership
Set.image
β€”mem_of_mem_image
Set.mem_image_of_mem
mem_of_mem_image πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.instMembership
Set.image
Set
Set.instMembership
β€”β€”
mono πŸ“–mathematicalSet
Set.instHasSubset
Set.InjOn
Set.InjOnβ€”β€”
ne πŸ“–β€”Set.InjOn
Set
Set.instMembership
β€”β€”ne_iff
ne_iff πŸ“–β€”Set.InjOn
Set
Set.instMembership
β€”β€”Iff.not
eq_iff
of_comp πŸ“–mathematicalSet.InjOnSet.InjOnβ€”β€”
preimage_image_inter πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set
Set.instInter
Set.preimage
Set.image
β€”Set.ext
mem_of_mem_image
Set.mem_image_of_mem
prodMap πŸ“–mathematicalSet.InjOnSet.InjOn
SProd.sprod
Set
Set.instSProd
β€”β€”
rightInvOn_of_leftInvOn πŸ“–mathematicalSet.InjOn
Set.LeftInvOn
Set.MapsTo
Set.RightInvOnβ€”β€”

Set.InvOn

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn πŸ“–mathematicalSet.InvOn
Set.MapsTo
Set.BijOnβ€”Set.LeftInvOn.injOn
Set.RightInvOn.surjOn
comp πŸ“–mathematicalSet.InvOn
Set.MapsTo
Set.InvOnβ€”Set.LeftInvOn.comp
Set.RightInvOn.comp
extendDomain πŸ“–mathematicalSet.InvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Set.InvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Set.LeftInvOn.extendDomain
Set.RightInvOn.extendDomain
mono πŸ“–mathematicalSet.InvOn
Set
Set.instHasSubset
Set.InvOnβ€”Set.LeftInvOn.mono
Set.RightInvOn.mono
prodMap πŸ“–mathematicalSet.InvOnSet.InvOn
SProd.sprod
Set
Set.instSProd
β€”Set.LeftInvOn.prodMap
Set.RightInvOn.prodMap

Set.LeftInvOn

Theorems

NameKindAssumesProvesValidatesDepends On
comp πŸ“–mathematicalSet.LeftInvOn
Set.MapsTo
Set.LeftInvOnβ€”β€”
congr_left πŸ“–mathematicalSet.LeftInvOn
Set.MapsTo
Set.EqOn
Set.LeftInvOnβ€”β€”
congr_right πŸ“–mathematicalSet.LeftInvOn
Set.EqOn
Set.LeftInvOnβ€”β€”
eq πŸ“–β€”Set.LeftInvOn
Set
Set.instMembership
β€”β€”β€”
eqOn πŸ“–mathematicalSet.LeftInvOnSet.EqOnβ€”β€”
extendDomain πŸ“–mathematicalSet.LeftInvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Set.LeftInvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Equiv.Perm.extendDomain_apply_image
image_image πŸ“–mathematicalSet.LeftInvOnSet.imageβ€”Set.image_image
Set.image_congr
Set.image_id'
image_image' πŸ“–mathematicalSet.LeftInvOn
Set
Set.instHasSubset
Set.imageβ€”image_image
mono
image_inter πŸ“–mathematicalSet.LeftInvOnSet.image
Set
Set.instInter
Set.preimage
β€”image_inter'
Set.Subset.antisymm
Set.mem_image_of_mem
Set.inter_subset_inter_left
Set.preimage_mono
Set.inter_subset_left
image_inter' πŸ“–mathematicalSet.LeftInvOnSet.image
Set
Set.instInter
Set.preimage
β€”Set.Subset.antisymm
Set.mem_preimage
Set.mem_image_of_mem
injOn πŸ“–mathematicalSet.LeftInvOnSet.InjOnβ€”β€”
mapsTo πŸ“–mathematicalSet.LeftInvOn
Set.SurjOn
Set.MapsToβ€”β€”
mono πŸ“–mathematicalSet.LeftInvOn
Set
Set.instHasSubset
Set.LeftInvOnβ€”β€”
prodMap πŸ“–mathematicalSet.LeftInvOnSet.LeftInvOn
SProd.sprod
Set
Set.instSProd
β€”β€”
rightInvOn_image πŸ“–mathematicalSet.LeftInvOnSet.RightInvOn
Set.image
β€”eq
surjOn πŸ“–mathematicalSet.LeftInvOn
Set.MapsTo
Set.SurjOnβ€”β€”

Set.MapsTo

Theorems

NameKindAssumesProvesValidatesDepends On
comp πŸ“–mathematicalSet.MapsToSet.MapsToβ€”β€”
comp_left πŸ“–mathematicalSet.MapsToSet.MapsTo
Set.image
β€”β€”
comp_right πŸ“–mathematicalSet.MapsToSet.MapsTo
Set.preimage
β€”β€”
extendDomain πŸ“–mathematicalSet.MapsTo
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Set.MapsTo
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Equiv.Perm.extendDomain_apply_image
image_subset πŸ“–mathematicalSet.MapsToSet
Set.instHasSubset
Set.image
β€”Set.mapsTo_iff_image_subset
insert πŸ“–mathematicalSet.MapsToSet.MapsTo
Set
Set.instInsert
β€”mono_right
Set.subset_union_right
inter πŸ“–mathematicalSet.MapsToSet.MapsTo
Set
Set.instInter
β€”β€”
inter_bijOn πŸ“–mathematicalSet.MapsTo
Set.BijOn
Set
Set.instHasSubset
Set.instInter
Set.preimage
Set.BijOn
Set
Set.instInter
β€”Set.BijOn.inter_mapsTo
Set.inter_comm
inter_inter πŸ“–mathematicalSet.MapsToSet.MapsTo
Set
Set.instInter
β€”β€”
iterate πŸ“–mathematicalSet.MapsToSet.MapsTo
Nat.iterate
β€”comp
iterate_restrict πŸ“–mathematicalSet.MapsToNat.iterate
Set.Elem
restrict
iterate
β€”iterate
coe_iterate_restrict
mem_iff πŸ“–mathematicalSet.MapsTo
Compl.compl
Set
Set.instCompl
Set
Set.instMembership
β€”by_contra
mono πŸ“–mathematicalSet.MapsTo
Set
Set.instHasSubset
Set.MapsToβ€”β€”
mono_left πŸ“–mathematicalSet.MapsTo
Set
Set.instHasSubset
Set.MapsToβ€”β€”
mono_right πŸ“–mathematicalSet.MapsTo
Set
Set.instHasSubset
Set.MapsToβ€”β€”
nonempty πŸ“–mathematicalSet.MapsTo
Set.Nonempty
Set.Nonemptyβ€”Set.Nonempty.mono
Set.mapsTo_iff_image_subset
Set.Nonempty.image
prodMap πŸ“–mathematicalSet.MapsToSet.MapsTo
SProd.sprod
Set
Set.instSProd
β€”β€”
subset_preimage πŸ“–mathematicalSet.MapsToSet
Set.instHasSubset
Set.preimage
β€”β€”
surjOn_compl πŸ“–mathematicalSet.MapsToSet.SurjOn
Compl.compl
Set
Set.instCompl
β€”Function.Surjective.forall
Set.mem_image_of_mem
union πŸ“–mathematicalSet.MapsToSet.MapsTo
Set
Set.instUnion
β€”union_union
Set.union_self
union_union πŸ“–mathematicalSet.MapsToSet.MapsTo
Set
Set.instUnion
β€”β€”

Set.RightInvOn

Theorems

NameKindAssumesProvesValidatesDepends On
comp πŸ“–mathematicalSet.RightInvOn
Set.MapsTo
Set.RightInvOnβ€”Set.LeftInvOn.comp
congr_left πŸ“–mathematicalSet.RightInvOn
Set.EqOn
Set.RightInvOnβ€”Set.LeftInvOn.congr_right
congr_right πŸ“–mathematicalSet.RightInvOn
Set.MapsTo
Set.EqOn
Set.RightInvOnβ€”Set.LeftInvOn.congr_left
eq πŸ“–β€”Set.RightInvOn
Set
Set.instMembership
β€”β€”β€”
eqOn πŸ“–mathematicalSet.RightInvOnSet.EqOnβ€”β€”
extendDomain πŸ“–mathematicalSet.RightInvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Set.RightInvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Equiv.Perm.extendDomain_apply_image
mapsTo πŸ“–mathematicalSet.RightInvOn
Set.SurjOn
Set.MapsToβ€”Set.LeftInvOn.mapsTo
mono πŸ“–mathematicalSet.RightInvOn
Set
Set.instHasSubset
Set.RightInvOnβ€”Set.LeftInvOn.mono
prodMap πŸ“–mathematicalSet.RightInvOnSet.RightInvOn
SProd.sprod
Set
Set.instSProd
β€”β€”
surjOn πŸ“–mathematicalSet.RightInvOn
Set.MapsTo
Set.SurjOnβ€”Set.LeftInvOn.surjOn

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
injOn πŸ“–mathematicalSet.SubsingletonSet.InjOnβ€”β€”

Set.SurjOn

Theorems

NameKindAssumesProvesValidatesDepends On
bijOn_subset πŸ“–mathematicalSet.SurjOnSet.BijOn
Set.image
Function.invFunOn
β€”Set.InvOn.bijOn
invOn_invFunOn
rightInvOn_invFunOn
Set.mapsTo_image
cancel_right πŸ“–mathematicalSet.SurjOn
Set.MapsTo
Set.EqOnβ€”Set.EqOn.cancel_right
Set.EqOn.comp_right
comap_nonempty πŸ“–mathematicalSet.SurjOn
Set.Nonempty
Set.Nonemptyβ€”Set.Nonempty.of_image
Set.Nonempty.mono
comp πŸ“–mathematicalSet.SurjOnSet.SurjOnβ€”Set.Subset.trans
Set.image_mono
Set.Subset.refl
Set.image_comp
comp_left πŸ“–mathematicalSet.SurjOnSet.SurjOn
Set.image
β€”eq_1
Set.image_comp
Set.image_mono
comp_right πŸ“–mathematicalSet.SurjOnSet.SurjOn
Set.preimage
β€”eq_1
Set.image_comp
Set.image_preimage_eq
exists_bijOn_subset πŸ“–mathematicalSet.SurjOnSet
Set.instHasSubset
Set.BijOn
β€”Set.surjOn_iff_exists_bijOn_subset
exists_subset_injOn_image_eq πŸ“–mathematicalSet.SurjOnSet
Set.instHasSubset
Set.InjOn
Set.image
β€”Set.ext
extendDomain πŸ“–mathematicalSet.SurjOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Set.SurjOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Equiv.Perm.extendDomain_apply_image
forall πŸ“–β€”Set.SurjOn
Set.MapsTo
β€”β€”β€”
image_eq_of_mapsTo πŸ“–mathematicalSet.SurjOn
Set.MapsTo
Set.imageβ€”Set.eq_of_subset_of_subset
Set.MapsTo.image_subset
image_invFunOn_image πŸ“–mathematicalSet.SurjOnSet.image
Function.invFunOn
β€”Set.LeftInvOn.image_image
rightInvOn_invFunOn
image_invFunOn_image_of_subset πŸ“–mathematicalSet.SurjOn
Set
Set.instHasSubset
Set.image
Function.invFunOn
β€”Set.LeftInvOn.image_image'
rightInvOn_invFunOn
image_preimage πŸ“–mathematicalSet.SurjOn
Set
Set.instHasSubset
Set.image
Set.preimage
β€”Set.image_preimage_eq_iff
Set.mem_range_of_mem_image
inter πŸ“–mathematicalSet.SurjOn
Set.InjOn
Set
Set.instUnion
Set.SurjOn
Set
Set.instInter
β€”inter_inter
Set.inter_self
inter_inter πŸ“–mathematicalSet.SurjOn
Set.InjOn
Set
Set.instUnion
Set.SurjOn
Set
Set.instInter
β€”Set.mem_image_of_mem
invOn_invFunOn πŸ“–mathematicalSet.SurjOnSet.InvOn
Function.invFunOn
Set.image
β€”rightInvOn_invFunOn
iterate πŸ“–mathematicalSet.SurjOnSet.SurjOn
Nat.iterate
β€”Set.surjOn_id
comp
leftInvOn_of_rightInvOn πŸ“–mathematicalSet.SurjOn
Set.RightInvOn
Set.LeftInvOnβ€”β€”
mapsTo_compl πŸ“–mathematicalSet.SurjOnSet.MapsTo
Compl.compl
Set
Set.instCompl
β€”β€”
mapsTo_invFunOn πŸ“–mathematicalSet.SurjOnSet.MapsTo
Function.invFunOn
β€”Set.mem_preimage
Function.invFunOn_mem
mono πŸ“–mathematicalSet
Set.instHasSubset
Set.SurjOn
Set.SurjOnβ€”Set.Subset.trans
Set.image_mono
of_comp πŸ“–mathematicalSet.SurjOn
Set.MapsTo
Set.SurjOnβ€”β€”
prodMap πŸ“–mathematicalSet.SurjOnSet.SurjOn
SProd.sprod
Set
Set.instSProd
β€”β€”
rightInvOn_invFunOn πŸ“–mathematicalSet.SurjOnSet.RightInvOn
Function.invFunOn
β€”Function.invFunOn_eq
subset_range πŸ“–mathematicalSet.SurjOnSet
Set.instHasSubset
Set.range
β€”Set.Subset.trans
Set.image_subset_range
surjective πŸ“–β€”Set.SurjOn
Set.univ
β€”β€”Set.surjOn_univ
mono
Set.subset_univ
Set.Subset.rfl
union πŸ“–mathematicalSet.SurjOnSet.SurjOn
Set
Set.instUnion
β€”β€”
union_union πŸ“–mathematicalSet.SurjOnSet.SurjOn
Set
Set.instUnion
β€”union
mono
Set.subset_union_left
Set.Subset.refl
Set.subset_union_right

Subtype

Theorems

NameKindAssumesProvesValidatesDepends On
coind_bijective πŸ“–mathematicalSet
Set.instMembership
Set.SurjOn
Set.univ
Function.Bijective
Set
Set.instMembership
coind
β€”coind_injective
coind_surjective
coind_surjective πŸ“–mathematicalSet
Set.instMembership
Set.SurjOn
Set.univ
Set
Set.instMembership
coind
β€”coe_injective

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