Documentation Verification Report

IsImage

📁 Source: Mathlib/Topology/OpenPartialHomeomorph/IsImage.lean

Statistics

MetricCount
DefinitionsIsImage, EqOnSource, IsImage, restr, eqOnSourceSetoid, ofSet, restr, restrOpen, IsImage
9
Theoremsrefl_toOpenPartialHomeomorph, refl_toPartialHomeomorph, symm_toOpenPartialHomeomorph, symm_toPartialHomeomorph, eqOn, restr, source_eq, symm', symm_eqOn_target, target_eq, apply_mem_iff, closure, compl, diff, frontier, iff_preimage_eq, iff_preimage_eq', iff_symm_preimage_eq, iff_symm_preimage_eq', image_eq, inter, inter_eq_of_inter_eq_of_eqOn, interior, isOpen_iff, leftInvOn_piecewise, map_nhdsWithin_eq, mapsTo, of_image_eq, of_preimage_eq, of_preimage_eq', of_symm_image_eq, of_symm_preimage_eq, of_symm_preimage_eq', preimage_eq, preimage_eq', restr_apply, restr_symm_apply, restr_toPartialEquiv, symm_apply_mem_iff, symm_eqOn_of_inter_eq_of_eqOn, symm_iff, symm_image_eq, symm_mapsTo, symm_preimage_eq, symm_preimage_eq', toPartialEquiv, union, restr_eqOn_source, coe_restrOpen, coe_restrOpen_symm, eqOnSource_iff, eqOnSource_refl, eq_of_eqOnSource_univ, isImage_source_target, isImage_source_target_of_disjoint, ofSet_apply, ofSet_source, ofSet_symm, ofSet_target, ofSet_toPartialEquiv, ofSet_univ_eq_refl, preimage_closure, preimage_frontier, preimage_interior, restrOpen_source, restrOpen_toPartialEquiv, restr_apply, restr_eqOnSource_restr, restr_eq_of_source_subset, restr_inter_source, restr_source, restr_source', restr_source_inter, restr_symm_apply, restr_target, restr_toPartialEquiv, restr_toPartialEquiv', restr_univ
78
Total87

CategoryTheory.Limits

Definitions

NameCategoryTheorems
IsImage 📖CompData

Homeomorph

Theorems

NameKindAssumesProvesValidatesDepends On
refl_toOpenPartialHomeomorph 📖mathematicaltoOpenPartialHomeomorph
refl
OpenPartialHomeomorph.refl
refl_toPartialHomeomorph 📖mathematicaltoOpenPartialHomeomorph
refl
OpenPartialHomeomorph.refl
refl_toOpenPartialHomeomorph
symm_toOpenPartialHomeomorph 📖mathematicaltoOpenPartialHomeomorph
symm
OpenPartialHomeomorph.symm
symm_toPartialHomeomorph 📖mathematicaltoOpenPartialHomeomorph
symm
OpenPartialHomeomorph.symm
symm_toOpenPartialHomeomorph

OpenPartialHomeomorph

Definitions

NameCategoryTheorems
EqOnSource 📖MathDef
2 mathmath: eqOnSource_iff, mem_contMDiffFiberwiseLinear_iff
IsImage 📖MathDef
14 mathmath: IsImage.of_symm_image_eq, IsImage.of_preimage_eq, Trivialization.isImage_preimage_prod, isImage_source_target_of_disjoint, IsImage.of_image_eq, IsImage.iff_preimage_eq', IsImage.of_symm_preimage_eq, isImage_source_target, IsImage.iff_symm_preimage_eq, IsImage.iff_symm_preimage_eq', IsImage.of_preimage_eq', IsImage.of_symm_preimage_eq', IsImage.symm_iff, IsImage.iff_preimage_eq
eqOnSourceSetoid 📖CompOp
9 mathmath: subtypeRestr_symm_trans_subtypeRestr, restr_inter_source, self_trans_symm, restr_eqOnSource_restr, symm_trans_self, eqOnSource_refl, symm_trans_restr, Set.EqOn.restr_eqOn_source, restr_symm_trans
ofSet 📖CompOp
15 mathmath: ofSet_toPartialEquiv, ofSet_source, ofSet_trans', self_trans_symm, symm_trans_self, ofSet_trans_ofSet, trans_ofSet, ofSet_symm, idRestrGroupoid_mem, ofSet_apply, trans_of_set', ofSet_mem_contDiffGroupoid, ofSet_target, ofSet_univ_eq_refl, ofSet_trans
restr 📖CompOp
26 mathmath: subtypeRestr_symm_trans_subtypeRestr, restr_inter_source, restr_target, EqOnSource.restr, ofSet_trans', restr_source_inter, restr_mem_maximalAtlas, restr_trans, restr_toPartialEquiv, ClosedUnderRestriction.closedUnderRestriction, restr_eqOnSource_restr, restr_source', restr_apply, trans_ofSet, restr_toPartialEquiv', symm_trans_restr, restr_source, restr_symm_apply, trans_of_set', restr_univ, Set.EqOn.restr_eqOn_source, Manifold.IsLocalSourceTargetProperty.mono_source, closedUnderRestriction', restr_eq_of_source_subset, ofSet_trans, restr_symm_trans
restrOpen 📖CompOp
4 mathmath: restrOpen_source, coe_restrOpen_symm, restrOpen_toPartialEquiv, coe_restrOpen

Theorems

NameKindAssumesProvesValidatesDepends On
coe_restrOpen 📖mathematicalIsOpentoFun'
restrOpen
coe_restrOpen_symm 📖mathematicalIsOpentoFun'
symm
restrOpen
eqOnSource_iff 📖mathematicalEqOnSource
PartialEquiv.EqOnSource
toPartialEquiv
eqOnSource_refl 📖mathematicalOpenPartialHomeomorph
eqOnSourceSetoid
eq_of_eqOnSource_univ 📖OpenPartialHomeomorph
eqOnSourceSetoid
PartialEquiv.source
toPartialEquiv
Set.univ
PartialEquiv.target
toPartialEquiv_injective
PartialEquiv.eq_of_eqOnSource_univ
isImage_source_target 📖mathematicalIsImage
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
PartialEquiv.isImage_source_target
isImage_source_target_of_disjoint 📖mathematicalDisjoint
Set
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
CompleteBooleanAlgebra.toCompleteLattice
CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra
Set.instCompleteAtomicBooleanAlgebra
HeytingAlgebra.toOrderBot
Order.Frame.toHeytingAlgebra
CompleteDistribLattice.toFrame
CompleteBooleanAlgebra.toCompleteDistribLattice
PartialEquiv.source
toPartialEquiv
PartialEquiv.target
IsImagePartialEquiv.isImage_source_target_of_disjoint
ofSet_apply 📖mathematicalIsOpentoFun'
ofSet
ofSet_source 📖mathematicalIsOpenPartialEquiv.source
toPartialEquiv
ofSet
ofSet_symm 📖mathematicalIsOpensymm
ofSet
ofSet_target 📖mathematicalIsOpenPartialEquiv.target
toPartialEquiv
ofSet
ofSet_toPartialEquiv 📖mathematicalIsOpentoPartialEquiv
ofSet
PartialEquiv.ofSet
ofSet_univ_eq_refl 📖mathematicalofSet
Set.univ
isOpen_univ
refl
ext
isOpen_univ
Set.ext
preimage_closure 📖mathematicalSet
Set.instInter
PartialEquiv.source
toPartialEquiv
Set.preimage
toFun'
closure
IsImage.preimage_eq
IsImage.closure
IsImage.of_preimage_eq
preimage_frontier 📖mathematicalSet
Set.instInter
PartialEquiv.source
toPartialEquiv
Set.preimage
toFun'
frontier
IsImage.preimage_eq
IsImage.frontier
IsImage.of_preimage_eq
preimage_interior 📖mathematicalSet
Set.instInter
PartialEquiv.source
toPartialEquiv
Set.preimage
toFun'
interior
IsImage.preimage_eq
IsImage.interior
IsImage.of_preimage_eq
restrOpen_source 📖mathematicalIsOpenPartialEquiv.source
toPartialEquiv
restrOpen
Set
Set.instInter
restrOpen_toPartialEquiv 📖mathematicalIsOpentoPartialEquiv
restrOpen
PartialEquiv.restr
restr_apply 📖mathematicaltoFun'
restr
restr_eqOnSource_restr 📖mathematicalSet
Set.instInter
PartialEquiv.source
toPartialEquiv
interior
OpenPartialHomeomorph
eqOnSourceSetoid
restr
restr_eq_of_source_subset 📖mathematicalSet
Set.instHasSubset
PartialEquiv.source
toPartialEquiv
restrtoPartialEquiv_injective
PartialEquiv.restr_eq_of_source_subset
interior_maximal
open_source
restr_inter_source 📖mathematicalOpenPartialHomeomorph
eqOnSourceSetoid
restr
Set
Set.instInter
PartialEquiv.source
toPartialEquiv
restr_eqOnSource_restr
interior_inter
interior_eq_iff_isOpen
open_source
restr_source 📖mathematicalPartialEquiv.source
toPartialEquiv
restr
Set
Set.instInter
interior
restr_source' 📖mathematicalIsOpenPartialEquiv.source
toPartialEquiv
restr
Set
Set.instInter
restr_source_inter 📖mathematicalrestr
Set
Set.instInter
PartialEquiv.source
toPartialEquiv
ext
interior_inter
IsOpen.interior_eq
open_source
Set.inter_self
restr_symm_apply 📖mathematicaltoFun'
symm
restr
restr_target 📖mathematicalPartialEquiv.target
toPartialEquiv
restr
Set
Set.instInter
Set.preimage
toFun'
symm
interior
restr_toPartialEquiv 📖mathematicaltoPartialEquiv
restr
PartialEquiv.restr
interior
restr_toPartialEquiv' 📖mathematicalIsOpentoPartialEquiv
restr
PartialEquiv.restr
restr_toPartialEquiv
IsOpen.interior_eq
restr_univ 📖mathematicalrestr
Set.univ
restr_eq_of_source_subset
Set.subset_univ

OpenPartialHomeomorph.EqOnSource

Theorems

NameKindAssumesProvesValidatesDepends On
eqOn 📖mathematicalOpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
Set.EqOn
OpenPartialHomeomorph.toFun'
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
restr 📖mathematicalOpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
OpenPartialHomeomorph.restrPartialEquiv.EqOnSource.restr
source_eq 📖mathematicalOpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
symm' 📖mathematicalOpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
OpenPartialHomeomorph.symmPartialEquiv.EqOnSource.symm'
symm_eqOn_target 📖mathematicalOpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
Set.EqOn
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
symm'
target_eq 📖mathematicalOpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
symm'

OpenPartialHomeomorph.IsImage

Definitions

NameCategoryTheorems
restr 📖CompOp
3 mathmath: restr_toPartialEquiv, restr_apply, restr_symm_apply

Theorems

NameKindAssumesProvesValidatesDepends On
apply_mem_iff 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instMembership
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
OpenPartialHomeomorph.toFun'
closure 📖mathematicalOpenPartialHomeomorph.IsImageclosuremap_nhdsWithin_eq
compl 📖mathematicalOpenPartialHomeomorph.IsImageCompl.compl
Set
Set.instCompl
Iff.not
diff 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instSDiff
inter
compl
frontier 📖mathematicalOpenPartialHomeomorph.IsImagefrontierdiff
closure
interior
iff_preimage_eq 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
PartialEquiv.IsImage.iff_preimage_eq
iff_preimage_eq' 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
PartialEquiv.target
symm_iff
iff_symm_preimage_eq'
iff_symm_preimage_eq 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
symm_iff
iff_preimage_eq
iff_symm_preimage_eq' 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
PartialEquiv.source
iff_symm_preimage_eq
OpenPartialHomeomorph.image_source_inter_eq
OpenPartialHomeomorph.image_source_inter_eq'
image_eq 📖mathematicalOpenPartialHomeomorph.IsImageSet.image
OpenPartialHomeomorph.toFun'
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.target
PartialEquiv.IsImage.image_eq
toPartialEquiv
inter 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instInter
Iff.and
inter_eq_of_inter_eq_of_eqOn 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.EqOn
OpenPartialHomeomorph.toFun'
PartialEquiv.targetPartialEquiv.IsImage.inter_eq_of_inter_eq_of_eqOn
toPartialEquiv
interior 📖mathematicalOpenPartialHomeomorph.IsImageinteriorclosure_compl
compl_compl
compl
closure
isOpen_iff 📖mathematicalOpenPartialHomeomorph.IsImageIsOpen
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.target
OpenPartialHomeomorph.isOpen_inter_preimage
symm_preimage_eq'
preimage_eq'
leftInvOn_piecewise 📖mathematicalOpenPartialHomeomorph.IsImageSet.LeftInvOn
Set.piecewise
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
Set.ite
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.IsImage.leftInvOn_piecewise
toPartialEquiv
map_nhdsWithin_eq 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instMembership
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Filter.map
OpenPartialHomeomorph.toFun'
nhdsWithin
OpenPartialHomeomorph.map_nhdsWithin_eq
image_eq
OpenPartialHomeomorph.nhdsWithin_target_inter
OpenPartialHomeomorph.map_source
mapsTo 📖mathematicalOpenPartialHomeomorph.IsImageSet.MapsTo
OpenPartialHomeomorph.toFun'
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.target
PartialEquiv.IsImage.mapsTo
toPartialEquiv
of_image_eq 📖mathematicalSet.image
OpenPartialHomeomorph.toFun'
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.target
OpenPartialHomeomorph.IsImagePartialEquiv.IsImage.of_image_eq
of_preimage_eq 📖mathematicalSet
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.IsImageiff_preimage_eq
of_preimage_eq' 📖mathematicalSet
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
PartialEquiv.target
OpenPartialHomeomorph.IsImageiff_preimage_eq'
of_symm_image_eq 📖mathematicalSet.image
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
Set
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.source
OpenPartialHomeomorph.IsImagePartialEquiv.IsImage.of_symm_image_eq
of_symm_preimage_eq 📖mathematicalSet
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
OpenPartialHomeomorph.IsImageiff_symm_preimage_eq
of_symm_preimage_eq' 📖mathematicalSet
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
PartialEquiv.source
OpenPartialHomeomorph.IsImageiff_symm_preimage_eq'
preimage_eq 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
iff_preimage_eq
preimage_eq' 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
PartialEquiv.target
iff_preimage_eq'
restr_apply 📖mathematicalOpenPartialHomeomorph.IsImage
IsOpen
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
OpenPartialHomeomorph.toFun'
restr
restr_symm_apply 📖mathematicalOpenPartialHomeomorph.IsImage
IsOpen
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
restr
restr_toPartialEquiv 📖mathematicalOpenPartialHomeomorph.IsImage
IsOpen
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
restr
PartialEquiv.IsImage.restr
toPartialEquiv
symm_apply_mem_iff 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instMembership
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
symm
symm_eqOn_of_inter_eq_of_eqOn 📖mathematicalOpenPartialHomeomorph.IsImage
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
Set.EqOn
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
PartialEquiv.target
PartialEquiv.IsImage.symm_eq_on_of_inter_eq_of_eqOn
toPartialEquiv
symm_iff 📖mathematicalOpenPartialHomeomorph.IsImage
OpenPartialHomeomorph.symm
symm
symm_image_eq 📖mathematicalOpenPartialHomeomorph.IsImageSet.image
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
Set
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.source
image_eq
symm
symm_mapsTo 📖mathematicalOpenPartialHomeomorph.IsImageSet.MapsTo
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
Set
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
PartialEquiv.source
mapsTo
symm
symm_preimage_eq 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
iff_symm_preimage_eq
symm_preimage_eq' 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instInter
PartialEquiv.target
OpenPartialHomeomorph.toPartialEquiv
Set.preimage
OpenPartialHomeomorph.toFun'
OpenPartialHomeomorph.symm
PartialEquiv.source
iff_symm_preimage_eq'
toPartialEquiv 📖mathematicalOpenPartialHomeomorph.IsImagePartialEquiv.IsImage
OpenPartialHomeomorph.toPartialEquiv
union 📖mathematicalOpenPartialHomeomorph.IsImageSet
Set.instUnion
Iff.or

OpenPartialHomeomorph.Set.EqOn

Theorems

NameKindAssumesProvesValidatesDepends On
restr_eqOn_source 📖mathematicalSet.EqOn
OpenPartialHomeomorph.toFun'
Set
Set.instInter
PartialEquiv.source
OpenPartialHomeomorph.toPartialEquiv
OpenPartialHomeomorph
OpenPartialHomeomorph.eqOnSourceSetoid
OpenPartialHomeomorph.restr
OpenPartialHomeomorph.restr_source'
OpenPartialHomeomorph.open_source
Set.inter_comm
Set.EqOn.trans

PartialEquiv

Definitions

NameCategoryTheorems
IsImage 📖MathDef
10 mathmath: IsImage.iff_symm_preimage_eq, IsImage.iff_preimage_eq, IsImage.of_symm_image_eq, isImage_source_target_of_disjoint, isImage_source_target, IsImage.of_preimage_eq, OpenPartialHomeomorph.IsImage.toPartialEquiv, IsImage.symm_iff, IsImage.of_image_eq, IsImage.of_symm_preimage_eq

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