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, 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', 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
279
Total280

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
Set.imageβ€”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.BijOnβ€”bijOn'
apply_symm_apply
bijOn' πŸ“–mathematicalSet.MapsTo
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
Set.BijOnβ€”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
19 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, 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
invFunOnβ€”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
invFunOnβ€”invFunOn_pos
invFunOn_neg πŸ“–mathematicalSet
Set.instMembership
invFunOnβ€”invFunOn.eq_1
invFunOn_pos πŸ“–mathematicalSet
Set.instMembership
invFunOnβ€”invFunOn.eq_1
leftInvOn_invFunOn_of_subset_image_image πŸ“–mathematicalSet
Set.instHasSubset
Set.image
invFunOn
Set.LeftInvOnβ€”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“–β€”Set.InjOnβ€”β€”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.imageβ€”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“–β€”Function.Semiconj
Set.InjOn
Set.image
β€”β€”Set.mem_image_of_mem
eq
injOn_preimage πŸ“–β€”Function.Semiconj
Set.InjOn
Set.preimage
β€”β€”eq
injOn_range πŸ“–β€”Function.Semiconj
Set.InjOn
Set.range
β€”β€”Set.image_univ
injOn_image
Function.Injective.injOn
mapsTo_image πŸ“–mathematicalFunction.Semiconj
Set.MapsTo
Set.imageβ€”β€”
mapsTo_image_right πŸ“–mathematicalFunction.Semiconj
Set.MapsTo
Set.imageβ€”Set.mapsTo_image_iff
mapsTo_preimage πŸ“–mathematicalFunction.Semiconj
Set.MapsTo
Set.preimageβ€”β€”
mapsTo_range πŸ“–mathematicalFunction.SemiconjSet.MapsTo
Set.range
β€”β€”
surjOn_image πŸ“–mathematicalFunction.Semiconj
Set.SurjOn
Set.imageβ€”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“–β€”BijOn
InjOn
image
β€”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€”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€”image
InjOn
β€”exists_subset_bijOn
BijOn.image_eq
BijOn.injOn
exists_image_eq_injOn_of_subset_range πŸ“–mathematicalSet
instHasSubset
range
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
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
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
instInsert
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
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
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' πŸ“–mathematicalβ€”MapsTo
Set
instHasSubset
image
β€”mapsTo_iff_image_subset
mapsTo_empty πŸ“–mathematicalβ€”MapsTo
Set
instEmptyCollection
β€”empty_subset
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€”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€”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
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“–β€”Set.BijOnβ€”β€”Set.MapsTo.comp
mapsTo
Set.InjOn.comp
injOn
Set.SurjOn.comp
surjOn
compl πŸ“–mathematicalSet.BijOn
Function.Bijective
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
Equiv.symmβ€”Equiv.bijOn_symm
exists πŸ“–mathematicalSet.BijOnSet
Set.instMembership
β€”surjOn
mapsTo
exists_extend πŸ“–β€”Set.BijOn
Set
Set.instHasSubset
Set.range
β€”β€”exists_extend_of_subset
Set.subset_univ
Set.image_univ
exists_extend_of_subset πŸ“–β€”Set.BijOn
Set
Set.instHasSubset
Set.SurjOn
β€”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
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.instInsertβ€”insert_iff
mapsTo
insert_iff πŸ“–mathematicalSet
Set.instMembership
Set.BijOn
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.instInterβ€”Set.MapsTo.inter_inter
mapsTo
Set.InjOn.mono
Set.inter_subset_left
injOn
Set.SurjOn.inter_inter
surjOn
inter_mapsTo πŸ“–β€”Set.BijOn
Set.MapsTo
Set
Set.instHasSubset
Set.instInter
Set.preimage
β€”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.BijOnNat.iterateβ€”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“–β€”Set.BijOn
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
β€”β€”Equiv.bijOn_symm
prodMap πŸ“–mathematicalSet.BijOnSProd.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.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.imageβ€”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.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“–β€”Set.BijOn
Set.InjOn
Set
Set.instUnion
β€”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“–β€”Set.EqOn
Set.InjOn
Set.MapsTo
β€”β€”β€”
cancel_right πŸ“–β€”Set.EqOn
Set.SurjOn
β€”β€”β€”
comp_eq πŸ“–β€”Set.EqOn
Set.range
β€”β€”Set.eqOn_range
comp_left πŸ“–β€”Set.EqOnβ€”β€”β€”
comp_leftβ‚‚ πŸ“–β€”Set.EqOnβ€”β€”β€”
comp_right πŸ“–β€”Set.EqOn
Set.MapsTo
β€”β€”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“–β€”Set
Set.instHasSubset
Set.EqOn
β€”β€”β€”
surjOn_iff πŸ“–mathematicalSet.EqOnSet.SurjOnβ€”Set.SurjOn.congr
symm
trans πŸ“–β€”Set.EqOnβ€”β€”β€”
union πŸ“–mathematicalSet.EqOnSet
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“–β€”Set.InjOn
Set.MapsTo
β€”β€”β€”
comp_iff πŸ“–mathematicalSet.InjOnSet.imageβ€”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.imageβ€”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
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.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.instInter
β€”Set.Subset.antisymm
Set.image_inter_subset
image_of_comp πŸ“–mathematicalSet.InjOnSet.imageβ€”Set.forall_mem_image
image_ssubset_image_iff πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.instHasSSubset
Set.image
β€”image_subset_image_iff
image_subset_image_iff πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.imageβ€”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
Nat.iterateβ€”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.imageβ€”mem_of_mem_image
Set.mem_image_of_mem
mem_of_mem_image πŸ“–β€”Set.InjOn
Set
Set.instHasSubset
Set.instMembership
Set.image
β€”β€”β€”
mono πŸ“–β€”Set
Set.instHasSubset
Set.InjOn
β€”β€”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“–β€”Set.InjOnβ€”β€”β€”
preimage_image_inter πŸ“–mathematicalSet.InjOn
Set
Set.instHasSubset
Set.instInter
Set.preimage
Set.image
β€”Set.ext
mem_of_mem_image
Set.mem_image_of_mem
prodMap πŸ“–mathematicalSet.InjOnSProd.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“–β€”Set.InvOn
Set.MapsTo
β€”β€”Set.LeftInvOn.comp
Set.RightInvOn.comp
extendDomain πŸ“–mathematicalSet.InvOn
DFunLike.coe
Equiv.Perm
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.Perm.extendDomain
Set.image
Equiv
β€”Set.LeftInvOn.extendDomain
Set.RightInvOn.extendDomain
mono πŸ“–β€”Set.InvOn
Set
Set.instHasSubset
β€”β€”Set.LeftInvOn.mono
Set.RightInvOn.mono
prodMap πŸ“–mathematicalSet.InvOnSProd.sprod
Set
Set.instSProd
β€”Set.LeftInvOn.prodMap
Set.RightInvOn.prodMap

Set.LeftInvOn

Theorems

NameKindAssumesProvesValidatesDepends On
comp πŸ“–β€”Set.LeftInvOn
Set.MapsTo
β€”β€”β€”
congr_left πŸ“–β€”Set.LeftInvOn
Set.MapsTo
Set.EqOn
β€”β€”β€”
congr_right πŸ“–β€”Set.LeftInvOn
Set.EqOn
β€”β€”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
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“–β€”Set.LeftInvOn
Set
Set.instHasSubset
β€”β€”β€”
prodMap πŸ“–mathematicalSet.LeftInvOnSProd.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“–β€”Set.MapsToβ€”β€”β€”
comp_left πŸ“–mathematicalSet.MapsToSet.imageβ€”β€”
comp_right πŸ“–mathematicalSet.MapsToSet.preimageβ€”β€”
extendDomain πŸ“–mathematicalSet.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
Set.instInsert
β€”mono_right
Set.subset_union_right
inter πŸ“–mathematicalSet.MapsToSet
Set.instInter
β€”β€”
inter_bijOn πŸ“–β€”Set.MapsTo
Set.BijOn
Set
Set.instHasSubset
Set.instInter
Set.preimage
β€”β€”Set.BijOn.inter_mapsTo
Set.inter_comm
inter_inter πŸ“–mathematicalSet.MapsToSet
Set.instInter
β€”β€”
iterate πŸ“–mathematicalSet.MapsToNat.iterateβ€”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.instMembershipβ€”by_contra
mono πŸ“–β€”Set.MapsTo
Set
Set.instHasSubset
β€”β€”β€”
mono_left πŸ“–β€”Set.MapsTo
Set
Set.instHasSubset
β€”β€”β€”
mono_right πŸ“–β€”Set.MapsTo
Set
Set.instHasSubset
β€”β€”β€”
nonempty πŸ“–β€”Set.MapsTo
Set.Nonempty
β€”β€”Set.Nonempty.mono
Set.mapsTo_iff_image_subset
Set.Nonempty.image
prodMap πŸ“–mathematicalSet.MapsToSProd.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
Set.instUnion
β€”union_union
Set.union_self
union_union πŸ“–mathematicalSet.MapsToSet
Set.instUnion
β€”β€”

Set.RightInvOn

Theorems

NameKindAssumesProvesValidatesDepends On
comp πŸ“–β€”Set.RightInvOn
Set.MapsTo
β€”β€”Set.LeftInvOn.comp
congr_left πŸ“–β€”Set.RightInvOn
Set.EqOn
β€”β€”Set.LeftInvOn.congr_right
congr_right πŸ“–β€”Set.RightInvOn
Set.MapsTo
Set.EqOn
β€”β€”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
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“–β€”Set.RightInvOn
Set
Set.instHasSubset
β€”β€”Set.LeftInvOn.mono
prodMap πŸ“–mathematicalSet.RightInvOnSProd.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“–β€”Set.SurjOn
Set.Nonempty
β€”β€”Set.Nonempty.of_image
Set.Nonempty.mono
comp πŸ“–β€”Set.SurjOnβ€”β€”Set.Subset.trans
Set.image_mono
Set.Subset.refl
Set.image_comp
comp_left πŸ“–mathematicalSet.SurjOnSet.imageβ€”eq_1
Set.image_comp
Set.image_mono
comp_right πŸ“–mathematicalSet.SurjOnSet.preimageβ€”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
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.instInterβ€”inter_inter
Set.inter_self
inter_inter πŸ“–mathematicalSet.SurjOn
Set.InjOn
Set
Set.instUnion
Set.instInterβ€”Set.mem_image_of_mem
invOn_invFunOn πŸ“–mathematicalSet.SurjOnSet.InvOn
Function.invFunOn
Set.image
β€”rightInvOn_invFunOn
iterate πŸ“–mathematicalSet.SurjOnNat.iterateβ€”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“–β€”Set
Set.instHasSubset
Set.SurjOn
β€”β€”Set.Subset.trans
Set.image_mono
of_comp πŸ“–β€”Set.SurjOn
Set.MapsTo
β€”β€”β€”
prodMap πŸ“–mathematicalSet.SurjOnSProd.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
Set.instUnion
β€”β€”
union_union πŸ“–mathematicalSet.SurjOnSet
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
coind
β€”coind_injective
coind_surjective
coind_surjective πŸ“–mathematicalSet
Set.instMembership
Set.SurjOn
Set.univ
coindβ€”coe_injective

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