Documentation Verification Report

Prod

📁 Source: Mathlib/Data/Set/Prod.lean

Statistics

MetricCount
Definitionsfst, snd, PullbackSelf, map_fst, map_snd, mapPullback, pullbackDiagonal, decidableMemDiagonal, decidableMemProd, toPullbackDiag
10
Theoremsset_prod, set_prod, piCongrLeft_preimage_pi, piCongrLeft_preimage_univ_pi, piCongrLeft_symm_preimage_pi, piCongrLeft_symm_preimage_univ_pi, sumPiEquivProdPi_symm_preimage_univ_pi, preimage_pullbackDiagonal, pullback_comm_sq, set_prod, set_prod, set_pi, set_prod_left, set_prod_right, left_of_eqOn_prodMap, prodMap, right_of_eqOn_prodMap, fst, graphOn, prod, snd, offDiag_nonempty, offDiag_eq_empty, prod, compl_prod_eq_union, diag_image, diag_preimage_prod, diag_preimage_prod_self, diagonal_eq_univ, diagonal_eq_univ_iff, diagonal_nonempty, diagonal_subset_iff, disjoint_diagonal_offDiag, disjoint_pi, disjoint_prod, disjoint_univ_pi, empty_pi, empty_prod, eqOn_prodMap_iff, eqOn_prod_iff, eval_image_pi, eval_image_pi_of_notMem, eval_image_pi_subset, eval_image_univ_pi, eval_image_univ_pi_subset, eval_preimage, eval_preimage', exists_eq_mgraphOn_univ, exists_equiv_range_eq_graphOn_univ, exists_prod_set, exists_range_eq_graphOn_univ, forall_prod_set, fst_image_prod, fst_image_prod_subset, fst_injOn_graph, graphOn_comp, graphOn_empty, graphOn_eq_empty, graphOn_inj, graphOn_insert, graphOn_nonempty, graphOn_prod_graphOn, graphOn_prod_prodMap, graphOn_singleton, graphOn_union, graphOn_univ_eq_range, image_fst_graphOn, image_prodMk_subset_prod, image_prodMk_subset_prod_left, image_prodMk_subset_prod_right, image_snd_graphOn, image_swap_prod, insert_pi, insert_prod, inter_prod, mapsTo_fst_prod, mapsTo_snd_prod, mapsTo_swap_prod, mem_graphOn, mk_preimage_prod, mk_preimage_prod_left, mk_preimage_prod_left_eq_empty, mk_preimage_prod_left_eq_if, mk_preimage_prod_left_fn_eq_if, mk_preimage_prod_right, mk_preimage_prod_right_eq_empty, mk_preimage_prod_right_eq_if, mk_preimage_prod_right_fn_eq_if, offDiag_empty, offDiag_eq_empty, offDiag_eq_sep_prod, offDiag_insert, offDiag_inter, offDiag_mono, offDiag_nonempty, offDiag_singleton, offDiag_subset_prod, offDiag_union, offDiag_univ, piMap_image_pi, piMap_image_pi_subset, piMap_image_univ_pi, piMap_mapsTo_pi, pi_congr, pi_eq_empty, pi_eq_empty_iff, pi_eq_empty_iff', pi_if, pi_inter_compl, pi_inter_distrib, pi_mono, pi_mono', pi_nonempty_iff, pi_subset_pi_iff, pi_univ, pi_univ_ite, pi_update_of_mem, pi_update_of_notMem, preimage_coe_coe_diagonal, preimage_pi, preimage_prod_map_prod, preimage_swap_prod, prodMap_image_prod, prod_diff_prod, prod_empty, prod_eq_empty_iff, prod_eq_iff_eq, prod_eq_prod_iff, prod_eq_prod_iff_of_nonempty, prod_eq_univ, prod_image_image_eq, prod_insert, prod_inter, prod_inter_prod, prod_mono, prod_mono_left, prod_mono_right, prod_nonempty_iff, prod_preimage_eq, prod_preimage_left, prod_preimage_right, prod_range_range_eq, prod_range_univ_eq, prod_sdiff_diagonal, prod_self_ssubset_prod_self, prod_self_subset_prod_self, prod_singleton, prod_sub_preimage_iff, prod_subset_compl_diagonal_iff_disjoint, prod_subset_iff, prod_subset_preimage_fst, prod_subset_preimage_snd, prod_subset_prod_iff, prod_subset_prod_iff', prod_subset_prod_iff_left, prod_subset_prod_iff_right, prod_subset_prod_left, prod_subset_prod_right, prod_union, prod_univ, prod_univ_range_eq, range_const_eq_diagonal, range_diag, range_pair_subset, range_piMap, range_prodMap, singleton_pi, singleton_pi', singleton_prod, singleton_prod_singleton, snd_image_prod, snd_image_prod_subset, subset_eval_image_pi, subset_fst_image_prod_snd_image, subset_pi_eval_image, subset_pi_iff, subset_prod, subsingleton_univ_pi, uncurry_preimage_prod_pi, union_pi, union_pi_inter, union_prod, uniqueElim_preimage, univ_pi_empty, univ_pi_eq_empty, univ_pi_eq_empty_iff, univ_pi_eq_singleton_iff, univ_pi_ite, univ_pi_nonempty_iff, univ_pi_singleton, univ_pi_subset_univ_pi_iff, univ_pi_update, univ_pi_update_univ, univ_prod, univ_prod_univ, update_image, update_mem_pi_iff, update_mem_pi_iff_of_mem, update_preimage_pi, update_preimage_univ_pi, image_toPullbackDiag, injective_toPullbackDiag, preimage_map_fst_pullbackDiagonal, range_toPullbackDiag, toPullbackDiag_coe
205
Total215

Antitone

Theorems

NameKindAssumesProvesValidatesDepends On
set_prod 📖mathematicalAntitone
Set
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
SProd.sprod
Set.instSProd
Set.prod_mono

AntitoneOn

Theorems

NameKindAssumesProvesValidatesDepends On
set_prod 📖mathematicalAntitoneOn
Set
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
SProd.sprod
Set.instSProd
Set.prod_mono

Equiv

Theorems

NameKindAssumesProvesValidatesDepends On
piCongrLeft_preimage_pi 📖mathematicalSet.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
piCongrLeft
Set.pi
Set.image
Set.ext
forall_congr_right
apply_symm_apply
piCongrLeft_symm_apply
piCongrLeft_preimage_univ_pi 📖mathematicalSet.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
piCongrLeft
Set.pi
Set.univ
Set.image_univ
Function.Surjective.range_eq
surjective
piCongrLeft_preimage_pi
piCongrLeft_symm_preimage_pi 📖mathematicalSet.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
piCongrLeft
Set.pi
Set.image
Set.ext
piCongrLeft_symm_apply
piCongrLeft_symm_preimage_univ_pi 📖mathematicalSet.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
piCongrLeft
Set.pi
Set.univ
Set.image_univ
Function.Surjective.range_eq
surjective
piCongrLeft_symm_preimage_pi
sumPiEquivProdPi_symm_preimage_univ_pi 📖mathematicalSet.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
sumPiEquivProdPi
Set.pi
Set.univ
SProd.sprod
Set
Set.instSProd
Set.ext

Function

Definitions

NameCategoryTheorems
PullbackSelf 📖CompOp
1 mathmath: preimage_map_fst_pullbackDiagonal
mapPullback 📖CompOp
2 mathmath: Injective.preimage_pullbackDiagonal, Continuous.mapPullback
pullbackDiagonal 📖CompOp
6 mathmath: isLocallyInjective_iff_isOpen_diagonal, range_toPullbackDiag, preimage_map_fst_pullbackDiagonal, image_toPullbackDiag, Injective.preimage_pullbackDiagonal, isSeparatedMap_iff_isClosed_diagonal

Theorems

NameKindAssumesProvesValidatesDepends On
pullback_comm_sq 📖mathematicalPullback
Pullback.fst
Pullback.snd

Function.Injective

Theorems

NameKindAssumesProvesValidatesDepends On
preimage_pullbackDiagonal 📖mathematicalSet.preimage
Function.Pullback
Function.mapPullback
Function.pullbackDiagonal
Set.ext

Function.Pullback

Definitions

NameCategoryTheorems
fst 📖CompOp
1 mathmath: Function.pullback_comm_sq
snd 📖CompOp
3 mathmath: IsSeparatedMap.pullback, Function.pullback_comm_sq, preimage_map_fst_pullbackDiagonal

Function.PullbackSelf

Definitions

NameCategoryTheorems
map_fst 📖CompOp
1 mathmath: preimage_map_fst_pullbackDiagonal
map_snd 📖CompOp

Monotone

Theorems

NameKindAssumesProvesValidatesDepends On
set_prod 📖mathematicalMonotone
Set
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
SProd.sprod
Set.instSProd
Set.prod_mono

MonotoneOn

Theorems

NameKindAssumesProvesValidatesDepends On
set_prod 📖mathematicalMonotoneOn
Set
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
SProd.sprod
Set.instSProd
Set.prod_mono

Set

Definitions

NameCategoryTheorems
decidableMemDiagonal 📖CompOp
decidableMemProd 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
compl_prod_eq_union 📖mathematicalCompl.compl
Set
instCompl
SProd.sprod
instSProd
instUnion
univ
diag_image 📖mathematicalimage
Set
instInter
diagonal
SProd.sprod
instSProd
range_diag
image_preimage_eq_range_inter
diag_preimage_prod_self
diag_preimage_prod 📖mathematicalpreimage
SProd.sprod
Set
instSProd
instInter
diag_preimage_prod_self 📖mathematicalpreimage
SProd.sprod
Set
instSProd
inter_self
diagonal_eq_univ 📖mathematicaldiagonal
univ
diagonal_eq_univ_iff
diagonal_eq_univ_iff 📖mathematicaldiagonal
univ
diagonal_nonempty 📖mathematicalNonempty
diagonal
mem_diagonal
diagonal_subset_iff 📖mathematicalSet
instHasSubset
diagonal
instMembership
range_diag
range_subset_iff
disjoint_diagonal_offDiag 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
diagonal
offDiag
disjoint_left
disjoint_pi 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
pi
instMembership
disjoint_prod 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
SProd.sprod
instSProd
disjoint_univ_pi 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
pi
univ
empty_pi 📖mathematicalpi
Set
instEmptyCollection
univ
empty_prod 📖mathematicalSProd.sprod
Set
instSProd
instEmptyCollection
ext
eqOn_prodMap_iff 📖mathematicalNonemptyEqOn
SProd.sprod
Set
instSProd
EqOn.left_of_eqOn_prodMap
EqOn.right_of_eqOn_prodMap
EqOn.prodMap
eqOn_prod_iff 📖mathematicalEqOn
eval_image_pi 📖mathematicalSet
instMembership
Nonempty
pi
image
Function.eval
HasSubset.Subset.antisymm
instAntisymmSubset
eval_image_pi_subset
subset_eval_image_pi
eval_image_pi_of_notMem 📖mathematicalSet
instMembership
image
Function.eval
pi
Nonempty
univ
instEmptyCollection
ext
image_congr
Function.update_of_ne
Function.update_self
eval_image_pi_subset 📖mathematicalSet
instMembership
instHasSubset
image
Function.eval
pi
image_subset_iff
eval_image_univ_pi 📖mathematicalNonempty
pi
univ
imageeval_image_pi
mem_univ
eval_image_univ_pi_subset 📖mathematicalSet
instHasSubset
image
Function.eval
pi
univ
eval_image_pi_subset
mem_univ
eval_preimage 📖mathematicalpreimage
Function.eval
pi
univ
Function.update
Set
ext
Function.forall_update_iff
eval_preimage' 📖mathematicalpreimage
Function.eval
pi
Set
instSingletonSet
Function.update
univ
ext
singleton_pi
Function.update_self
exists_eq_mgraphOn_univ 📖mathematicalFunction.Bijective
Set
instMembership
graphOn
univ
Subtype.range_coe_subtype
exists_range_eq_graphOn_univ
Function.Bijective.surjective
Function.Bijective.injective
exists_equiv_range_eq_graphOn_univ 📖mathematicalrange
graphOn
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
univ
exists_range_eq_graphOn_univ
ext_iff
exists_prod_set 📖mathematicalSet
instMembership
SProd.sprod
instSProd
exists_range_eq_graphOn_univ 📖mathematicalrange
graphOn
univ
ext
forall_prod_set 📖prod_subset_iff
fst_image_prod 📖mathematicalNonemptyimage
SProd.sprod
Set
instSProd
HasSubset.Subset.antisymm
instAntisymmSubset
fst_image_prod_subset
fst_image_prod_subset 📖mathematicalSet
instHasSubset
image
SProd.sprod
instSProd
image_subset_iff
prod_subset_preimage_fst
fst_injOn_graph 📖mathematicalInjOn
graphOn
graphOn_comp 📖mathematicalgraphOn
image
image_congr
image_comp
graphOn_empty 📖mathematicalgraphOn
Set
instEmptyCollection
image_empty
graphOn_eq_empty 📖mathematicalgraphOn
Set
instEmptyCollection
image_eq_empty
graphOn_inj 📖mathematicalgraphOn
EqOn
graphOn_insert 📖mathematicalgraphOn
Set
instInsert
image_insert_eq
graphOn_nonempty 📖mathematicalNonempty
graphOn
image_nonempty
graphOn_prod_graphOn 📖mathematicalSProd.sprod
Set
instSProd
graphOn
preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.prodProdProdComm
ext
graphOn_prod_prodMap 📖mathematicalgraphOn
SProd.sprod
Set
instSProd
preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.prodProdProdComm
ext
graphOn_singleton 📖mathematicalgraphOn
Set
instSingletonSet
image_singleton
graphOn_union 📖mathematicalgraphOn
Set
instUnion
image_union
graphOn_univ_eq_range 📖mathematicalgraphOn
univ
range
image_univ
image_fst_graphOn 📖mathematicalimage
graphOn
image_image
image_congr
image_id'
image_prodMk_subset_prod 📖mathematicalSet
instHasSubset
image
SProd.sprod
instSProd
image_prodMk_subset_prod_left 📖mathematicalSet
instMembership
instHasSubset
image
SProd.sprod
instSProd
image_prodMk_subset_prod_right 📖mathematicalSet
instMembership
instHasSubset
image
SProd.sprod
instSProd
image_snd_graphOn 📖mathematicalimage
graphOn
ext
image_swap_prod 📖mathematicalimage
SProd.sprod
Set
instSProd
image_swap_eq_preimage_swap
preimage_swap_prod
insert_pi 📖mathematicalpi
Set
instInsert
instInter
preimage
Function.eval
insert_prod 📖mathematicalSProd.sprod
Set
instSProd
instInsert
instUnion
image
union_prod
singleton_prod
inter_prod 📖mathematicalSProd.sprod
Set
instSProd
instInter
ext
mapsTo_fst_prod 📖mathematicalMapsTo
SProd.sprod
Set
instSProd
mem_prod
mapsTo_snd_prod 📖mathematicalMapsTo
SProd.sprod
Set
instSProd
mem_prod
mapsTo_swap_prod 📖mathematicalMapsTo
SProd.sprod
Set
instSProd
mem_graphOn 📖mathematicalSet
instMembership
graphOn
mk_preimage_prod 📖mathematicalpreimage
SProd.sprod
Set
instSProd
instInter
mk_preimage_prod_left 📖mathematicalSet
instMembership
preimage
SProd.sprod
instSProd
mk_preimage_prod_left_eq_empty 📖mathematicalSet
instMembership
preimage
SProd.sprod
instSProd
instEmptyCollection
mk_preimage_prod_left_eq_if 📖mathematicalpreimage
SProd.sprod
Set
instSProd
instMembership
instEmptyCollection
mk_preimage_prod_left_fn_eq_if 📖mathematicalpreimage
SProd.sprod
Set
instSProd
instMembership
instEmptyCollection
mk_preimage_prod_right 📖mathematicalSet
instMembership
preimage
SProd.sprod
instSProd
mk_preimage_prod_right_eq_empty 📖mathematicalSet
instMembership
preimage
SProd.sprod
instSProd
instEmptyCollection
mk_preimage_prod_right_eq_if 📖mathematicalpreimage
SProd.sprod
Set
instSProd
instMembership
instEmptyCollection
mk_preimage_prod_right_fn_eq_if 📖mathematicalpreimage
SProd.sprod
Set
instSProd
instMembership
instEmptyCollection
offDiag_empty 📖mathematicaloffDiag
Set
instEmptyCollection
offDiag_eq_empty 📖mathematicaloffDiag
Set
instEmptyCollection
Subsingleton
not_nonempty_iff_eq_empty
not_nontrivial_iff
Iff.not
offDiag_nonempty
offDiag_eq_sep_prod 📖mathematicaloffDiag
setOf
Set
instMembership
SProd.sprod
instSProd
ext
offDiag_insert 📖mathematicalSet
instMembership
offDiag
instInsert
instUnion
SProd.sprod
instSProd
instSingletonSet
offDiag_inter 📖mathematicaloffDiag
Set
instInter
ext
offDiag_mono 📖mathematicalMonotone
Set
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
offDiag
offDiag_nonempty 📖mathematicalNonempty
offDiag
Nontrivial
offDiag_singleton 📖mathematicaloffDiag
Set
instSingletonSet
instEmptyCollection
offDiag_subset_prod 📖mathematicalSet
instHasSubset
offDiag
SProd.sprod
instSProd
offDiag_union 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
offDiag
instUnion
SProd.sprod
instSProd
ext
disjoint_left
disjoint_right
offDiag_univ 📖mathematicaloffDiag
univ
Compl.compl
Set
instCompl
diagonal
ext
piMap_image_pi 📖mathematicalimage
Pi.map
pi
Subset.antisymm
piMap_image_pi_subset
piMap_image_pi_subset 📖mathematicalSet
instHasSubset
image
Pi.map
pi
image_subset_iff
piMap_mapsTo_pi
mapsTo_image
piMap_image_univ_pi 📖mathematicalimage
Pi.map
pi
univ
piMap_image_pi
instIsEmptyFalse
piMap_mapsTo_pi 📖mathematicalMapsToPi.map
pi
pi_congr 📖mathematicalpi
pi_eq_empty 📖mathematicalSet
instMembership
instEmptyCollection
pi
pi_eq_empty_iff 📖mathematicalpi
Set
instEmptyCollection
IsEmpty
instMembership
not_nonempty_iff_eq_empty
pi_nonempty_iff
Mathlib.Tactic.Push.not_forall_eq
isEmpty_or_nonempty
pi_eq_empty_iff' 📖mathematicalpi
Set
instEmptyCollection
instMembership
pi_if 📖mathematicalpi
Set
instInter
setOf
instMembership
ext
pi_inter_compl 📖mathematicalSet
instInter
pi
Compl.compl
instCompl
univ
pi_inter_distrib 📖mathematicalpi
Set
instInter
pi_mono 📖mathematicalSet
instHasSubset
pipi_mono'
Subset.rfl
pi_mono' 📖mathematicalSet
instHasSubset
pi
pi_nonempty_iff 📖mathematicalNonempty
pi
Set
instMembership
pi_subset_pi_iff 📖mathematicalSet
instHasSubset
pi
instEmptyCollection
nonempty_iff_ne_empty
eval_image_pi
Nonempty.mono
image_mono
pi_mono
empty_subset
pi_univ 📖mathematicalpi
univ
eq_univ_of_forall
mem_univ
pi_univ_ite 📖mathematicalpi
univ
Set
instMembership
pi_update_of_mem 📖mathematicalSet
instMembership
pi
Function.update
instInter
setOf
instSDiff
instSingletonSet
union_diff_self
union_eq_self_of_subset_left
singleton_subset_iff
union_pi
singleton_pi'
Function.update_self
pi_update_of_notMem
pi_update_of_notMem 📖mathematicalSet
instMembership
pi
Function.update
pi_congr
Function.update_of_ne
preimage_coe_coe_diagonal 📖mathematicalpreimage
Elem
Set
instMembership
diagonal
ext
preimage_pi 📖mathematicalpreimage
pi
preimage_prod_map_prod 📖mathematicalpreimage
SProd.sprod
Set
instSProd
preimage_swap_prod 📖mathematicalpreimage
SProd.sprod
Set
instSProd
prodMap_image_prod 📖mathematicalimage
SProd.sprod
Set
instSProd
ext
prod_diff_prod 📖mathematicalSet
instSDiff
SProd.sprod
instSProd
instUnion
prod_empty 📖mathematicalSProd.sprod
Set
instSProd
instEmptyCollection
ext
prod_eq_empty_iff 📖mathematicalSProd.sprod
Set
instSProd
instEmptyCollection
not_nonempty_iff_eq_empty
prod_eq_iff_eq 📖mathematicalNonemptySProd.sprod
Set
instSProd
Nonempty.ne_empty
prod_eq_prod_iff 📖mathematicalSProd.sprod
Set
instSProd
instEmptyCollection
eq_empty_or_nonempty
prod_eq_empty_iff
prod_eq_prod_iff_of_nonempty
nonempty_iff_ne_empty
prod_eq_prod_iff_of_nonempty 📖Nonempty
SProd.sprod
Set
instSProd
fst_image_prod
prod_nonempty_iff
snd_image_prod
prod_eq_univ 📖mathematicalSProd.sprod
Set
instSProd
univ
prod_image_image_eq 📖mathematicalSProd.sprod
Set
instSProd
image
ext
prod_insert 📖mathematicalSProd.sprod
Set
instSProd
instInsert
instUnion
image
prod_union
prod_singleton
prod_inter 📖mathematicalSProd.sprod
Set
instSProd
instInter
ext
prod_inter_prod 📖mathematicalSet
instInter
SProd.sprod
instSProd
ext
prod_mono 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
prod_mono_left 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
prod_mono
Subset.rfl
prod_mono_right 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
prod_mono
Subset.rfl
prod_nonempty_iff 📖mathematicalNonempty
SProd.sprod
Set
instSProd
Nonempty.fst
Nonempty.snd
Nonempty.prod
prod_preimage_eq 📖mathematicalSProd.sprod
Set
instSProd
preimage
prod_preimage_left 📖mathematicalSProd.sprod
Set
instSProd
preimage
prod_preimage_right 📖mathematicalSProd.sprod
Set
instSProd
preimage
prod_range_range_eq 📖mathematicalSProd.sprod
Set
instSProd
range
ext
prod_range_univ_eq 📖mathematicalSProd.sprod
Set
instSProd
range
univ
ext
prod_sdiff_diagonal 📖mathematicalSet
instSDiff
SProd.sprod
instSProd
diagonal
offDiag
ext
prod_self_ssubset_prod_self 📖mathematicalSet
instHasSSubset
SProd.sprod
instSProd
prod_self_subset_prod_self
prod_self_subset_prod_self 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
mk_mem_prod
prod_singleton 📖mathematicalSProd.sprod
Set
instSProd
instSingletonSet
image
ext
prod_sub_preimage_iff 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
preimage
instMembership
prod_subset_compl_diagonal_iff_disjoint 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
Compl.compl
instCompl
diagonal
Disjoint
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
prod_subset_iff
disjoint_iff_forall_ne
prod_subset_iff 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
instMembership
mk_mem_prod
prod_subset_preimage_fst 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
preimage
inter_subset_left
prod_subset_preimage_snd 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
preimage
inter_subset_right
prod_subset_prod_iff 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
instEmptyCollection
eq_empty_or_nonempty
prod_eq_empty_iff
prod_nonempty_iff
image_mono
fst_image_prod
Nonempty.snd
Nonempty.mono
snd_image_prod
Nonempty.fst
prod_mono
Nonempty.ne_empty
prod_subset_prod_iff' 📖mathematicalNonempty
SProd.sprod
Set
instSProd
instHasSubsetprod_subset_prod_iff
prod_eq_empty_iff
Nonempty.ne_empty
prod_subset_prod_iff_left 📖mathematicalNonemptySet
instHasSubset
SProd.sprod
instSProd
instReflSubset
Nonempty.ne_empty
prod_subset_prod_iff_right 📖mathematicalNonemptySet
instHasSubset
SProd.sprod
instSProd
instReflSubset
Nonempty.ne_empty
prod_subset_prod_left 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
prod_mono_left
prod_subset_prod_right 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
prod_mono_right
prod_union 📖mathematicalSProd.sprod
Set
instSProd
instUnion
ext
prod_univ 📖mathematicalSProd.sprod
Set
instSProd
univ
preimage
inter_univ
prod_univ_range_eq 📖mathematicalSProd.sprod
Set
instSProd
univ
range
ext
range_const_eq_diagonal 📖mathematicalrange
setOf
range_eq_iff
isEmpty_or_nonempty
Unique.instSubsingleton
range_diag 📖mathematicalrange
diagonal
ext
range_pair_subset 📖mathematicalSet
instHasSubset
range
SProd.sprod
instSProd
range_piMap 📖mathematicalrange
Pi.map
pi
univ
pi_univ
range_prodMap 📖mathematicalrange
SProd.sprod
Set
instSProd
prod_range_range_eq
singleton_pi 📖mathematicalpi
Set
instSingletonSet
preimage
Function.eval
singleton_pi' 📖mathematicalpi
Set
instSingletonSet
setOf
instMembership
singleton_pi
singleton_prod 📖mathematicalSProd.sprod
Set
instSProd
instSingletonSet
image
ext
singleton_prod_singleton 📖mathematicalSProd.sprod
Set
instSProd
instSingletonSet
ext
snd_image_prod 📖mathematicalNonemptyimage
SProd.sprod
Set
instSProd
HasSubset.Subset.antisymm
instAntisymmSubset
snd_image_prod_subset
snd_image_prod_subset 📖mathematicalSet
instHasSubset
image
SProd.sprod
instSProd
image_subset_iff
prod_subset_preimage_snd
subset_eval_image_pi 📖mathematicalNonempty
pi
Set
instHasSubset
image
Function.eval
eq_or_ne
Function.update_self
Function.update_of_ne
subset_fst_image_prod_snd_image 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
image
subset_pi_eval_image 📖mathematicalSet
instHasSubset
pi
image
Function.eval
subset_pi_iff 📖mathematicalSet
instHasSubset
pi
preimage
subset_prod 📖mathematicalSet
instHasSubset
SProd.sprod
instSProd
image
mem_prod
mem_image_of_mem
subsingleton_univ_pi 📖mathematicalSubsingletonpi
univ
mem_univ
uncurry_preimage_prod_pi 📖mathematicalpreimage
pi
SProd.sprod
Set
instSProd
union_pi 📖mathematicalpi
Set
instUnion
instInter
union_pi_inter 📖mathematicalunivpi
Set
instUnion
instInter
union_prod 📖mathematicalSProd.sprod
Set
instSProd
instUnion
ext
uniqueElim_preimage 📖mathematicalpreimage
Unique.instInhabited
uniqueElim
pi
univ
ext
univ_pi_empty 📖mathematicalpi
univ
Set
instEmptyCollection
univ_pi_eq_empty_iff
univ_pi_eq_empty 📖mathematicalSet
instEmptyCollection
pi
univ
pi_eq_empty
mem_univ
univ_pi_eq_empty_iff 📖mathematicalpi
univ
Set
instEmptyCollection
univ_pi_eq_singleton_iff 📖mathematicalpi
univ
Set
instSingletonSet
update_mem_pi_iff_of_mem
Function.update_self
univ_pi_ite 📖mathematicalpi
univ
Set
instMembership
univ_pi_nonempty_iff 📖mathematicalNonempty
pi
univ
univ_pi_singleton 📖mathematicalpi
univ
Set
instSingletonSet
ext
univ_pi_subset_univ_pi_iff 📖mathematicalSet
instHasSubset
pi
univ
instEmptyCollection
univ_pi_update 📖mathematicalpi
univ
Function.update
Set
instInter
setOf
instMembership
Compl.compl
instCompl
instSingletonSet
compl_eq_univ_diff
pi_update_of_mem
mem_univ
univ_pi_update_univ 📖mathematicalpi
univ
Function.update
Set
preimage
Function.eval
univ_pi_update
pi_univ
inter_univ
preimage.eq_1
univ_prod 📖mathematicalSProd.sprod
Set
instSProd
univ
preimage
univ_inter
univ_prod_univ 📖mathematicalSProd.sprod
Set
instSProd
univ
ext
update_image 📖mathematicalimage
Function.update
pi
univ
Set
instSingletonSet
ext
Function.forall_update_iff
update_mem_pi_iff 📖mathematicalSet
instMembership
pi
Function.update
instSDiff
instSingletonSet
update_mem_pi_iff_of_mem 📖mathematicalSet
instMembership
pi
Function.updateupdate_mem_pi_iff
update_preimage_pi 📖mathematicalSet
instMembership
preimage
Function.update
pi
ext
Function.update_self
eq_or_ne
Function.update_of_ne
update_preimage_univ_pi 📖mathematicalSet
instMembership
preimage
Function.update
pi
univ
update_preimage_pi
mem_univ

Set.Disjoint

Theorems

NameKindAssumesProvesValidatesDepends On
set_pi 📖mathematicalSet
Set.instMembership
Disjoint
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
Set.piSet.disjoint_left
set_prod_left 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
SProd.sprod
Set.instSProd
Set.disjoint_left
set_prod_right 📖mathematicalDisjoint
Set
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
GeneralizedCoheytingAlgebra.toLattice
CoheytingAlgebra.toGeneralizedCoheytingAlgebra
BiheytingAlgebra.toCoheytingAlgebra
BooleanAlgebra.toBiheytingAlgebra
Set.instBooleanAlgebra
HeytingAlgebra.toOrderBot
BiheytingAlgebra.toHeytingAlgebra
SProd.sprod
Set.instSProd
Set.disjoint_left

Set.EqOn

Theorems

NameKindAssumesProvesValidatesDepends On
left_of_eqOn_prodMap 📖Set.EqOn
SProd.sprod
Set
Set.instSProd
Set.Nonempty
Set.mk_mem_prod
prodMap 📖mathematicalSet.EqOnSProd.sprod
Set
Set.instSProd
right_of_eqOn_prodMap 📖Set.EqOn
SProd.sprod
Set
Set.instSProd
Set.Nonempty
Set.mk_mem_prod

Set.Nonempty

Theorems

NameKindAssumesProvesValidatesDepends On
fst 📖Set.Nonempty
SProd.sprod
Set
Set.instSProd
graphOn 📖mathematicalSet.NonemptySet.graphOnSet.graphOn_nonempty
prod 📖mathematicalSet.NonemptySProd.sprod
Set
Set.instSProd
snd 📖Set.Nonempty
SProd.sprod
Set
Set.instSProd

Set.Nontrivial

Theorems

NameKindAssumesProvesValidatesDepends On
offDiag_nonempty 📖mathematicalSet.NontrivialSet.Nonempty
Set.offDiag
Set.offDiag_nonempty

Set.Subsingleton

Theorems

NameKindAssumesProvesValidatesDepends On
offDiag_eq_empty 📖mathematicalSet.SubsingletonSet.offDiag
Set
Set.instEmptyCollection
Set.offDiag_eq_empty
prod 📖mathematicalSet.SubsingletonSProd.sprod
Set
Set.instSProd

(root)

Definitions

NameCategoryTheorems
toPullbackDiag 📖CompOp
9 mathmath: injective_toPullbackDiag, toPullbackDiag_coe, isSeparatedMap_iff_isClosedEmbedding, isSeparatedMap_iff_isClosedMap, isLocallyInjective_iff_isOpenMap, Topology.IsEmbedding.toPullbackDiag, range_toPullbackDiag, IsLocallyInjective_iff_isOpenEmbedding, image_toPullbackDiag

Theorems

NameKindAssumesProvesValidatesDepends On
image_toPullbackDiag 📖mathematicalSet.image
Function.Pullback
toPullbackDiag
Set
Set.instInter
Function.pullbackDiagonal
Set.preimage
SProd.sprod
Set.instSProd
Set.ext
Set.mem_image_of_mem
injective_toPullbackDiag 📖mathematicalFunction.Pullback
toPullbackDiag
preimage_map_fst_pullbackDiagonal 📖mathematicalSet.preimage
Function.PullbackSelf
Function.Pullback
Function.Pullback.snd
Function.PullbackSelf.map_fst
Function.pullbackDiagonal
Set.ext
range_toPullbackDiag 📖mathematicalSet.range
Function.Pullback
toPullbackDiag
Function.pullbackDiagonal
Set.image_univ
image_toPullbackDiag
Set.univ_prod_univ
Set.preimage_univ
Set.inter_univ
toPullbackDiag_coe 📖mathematicaltoPullbackDiag

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