ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PartialHomeomorph.source_inter_preimage_target_inter
theorem source_inter_preimage_target_inter (s : Set Y) :
    e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set Y)
Goal: e.source ∩ ↑e ⁻¹' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' s
-- PartialHomeomorph.image_source_eq_target
theorem image_source_eq_target : e '' e.source = e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ↑e '' e.source = e.target
-- PartialHomeomorph.symm_image_target_eq_source
theorem symm_image_target_eq_source : e.symm '' e.target = e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ↑e.symm '' e.target = e.source
-- PartialHomeomorph.ext
@[ext]
protected theorem ext (e' : PartialHomeomorph X Y) (h : ∀ x, e x = e' x)
    (hinv : ∀ x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e e' : PartialHomeomorph X Y) (h : ∀ (x : X), ↑e x = ↑e' x) (hinv : ∀ (x : Y), ↑e.symm x = ↑e'.symm x) (hs : e.source = e'.source)
Goal: e = e'
-- PartialHomeomorph.ext_iff
protected theorem ext_iff {e e' : PartialHomeomorph X Y} :
    e = e' ↔ (∀ x, e x = e' x) ∧ (∀ x, e.symm x = e'.symm x) ∧ e.source = e'.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e e' : PartialHomeomorph X Y}
Goal: e = e' ↔ (∀ (x : X), ↑e x = ↑e' x) ∧ (∀ (x : Y), ↑e.symm x = ↑e'.symm x) ∧ e.source = e'.source
-- PartialHomeomorph.symm_toPartialEquiv
@[simp, mfld_simps]
theorem symm_toPartialEquiv : e.symm.toPartialEquiv = e.toPartialEquiv.symm 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.symm.toPartialEquiv = e.symm
-- PartialHomeomorph.symm_source
theorem symm_source : e.symm.source = e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.symm.source = e.target
-- PartialHomeomorph.symm_target
theorem symm_target : e.symm.target = e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.symm.target = e.source
-- PartialHomeomorph.symm_symm
@[simp, mfld_simps] theorem symm_symm : e.symm.symm = e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.symm.symm = e
-- PartialHomeomorph.symm_bijective
theorem symm_bijective : Function.Bijective
    (PartialHomeomorph.symm : PartialHomeomorph X Y → PartialHomeomorph Y X) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Function.Bijective PartialHomeomorph.symm
-- PartialHomeomorph.continuousAt
protected theorem continuousAt {x : X} (h : x ∈ e.source) : ContinuousAt e x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (h : x ∈ e.source)
Goal: ContinuousAt (↑e) x
-- PartialHomeomorph.continuousAt_symm
theorem continuousAt_symm {x : Y} (h : x ∈ e.target) : ContinuousAt e.symm x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : Y} (h : x ∈ e.target)
Goal: ContinuousAt (↑e.symm) x
-- PartialHomeomorph.tendsto_symm
theorem tendsto_symm {x} (hx : x ∈ e.source) : Tendsto e.symm (𝓝 (e x)) (𝓝 x) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: Filter.Tendsto (↑e.symm) (𝓝 (↑e x)) (𝓝 x)
-- PartialHomeomorph.map_nhds_eq
theorem map_nhds_eq {x} (hx : x ∈ e.source) : map e (𝓝 x) = 𝓝 (e x) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: Filter.map (↑e) (𝓝 x) = 𝓝 (↑e x)
-- PartialHomeomorph.symm_map_nhds_eq
theorem symm_map_nhds_eq {x} (hx : x ∈ e.source) : map e.symm (𝓝 (e x)) = 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: Filter.map (↑e.symm) (𝓝 (↑e x)) = 𝓝 x
-- PartialHomeomorph.image_mem_nhds
theorem image_mem_nhds {x} (hx : x ∈ e.source) {s : Set X} (hs : s ∈ 𝓝 x) : e '' s ∈ 𝓝 (e x) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source) {s : Set X} (hs : s ∈ 𝓝 x)
Goal: ↑e '' s ∈ 𝓝 (↑e x)
-- PartialHomeomorph.map_nhdsWithin_eq
theorem map_nhdsWithin_eq {x} (hx : x ∈ e.source) (s : Set X) :
    map e (𝓝[s] x) = 𝓝[e '' (e.source ∩ s)] e x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source) (s : Set X)
Goal: Filter.map (↑e) (𝓝[s] x) = 𝓝[↑e '' (e.source ∩ s)] ↑e x
-- PartialHomeomorph.map_nhdsWithin_preimage_eq
theorem map_nhdsWithin_preimage_eq {x} (hx : x ∈ e.source) (s : Set Y) :
    map e (𝓝[e ⁻¹' s] x) = 𝓝[s] e x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source) (s : Set Y)
Goal: Filter.map (↑e) (𝓝[↑e ⁻¹' s] x) = 𝓝[s] ↑e x
-- PartialHomeomorph.eventually_nhds
theorem eventually_nhds {x : X} (p : Y → Prop) (hx : x ∈ e.source) :
    (∀ᶠ y in 𝓝 (e x), p y) ↔ ∀ᶠ x in 𝓝 x, p (e x) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (p : Y → Prop) (hx : x ∈ e.source)
Goal: (∀ᶠ (y : Y) in 𝓝 (↑e x), p y) ↔ ∀ᶠ (x : X) in 𝓝 x, p (↑e x)
-- PartialHomeomorph.eventually_nhds'
theorem eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) :
    (∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (p : X → Prop) (hx : x ∈ e.source)
Goal: (∀ᶠ (y : Y) in 𝓝 (↑e x), p (↑e.symm y)) ↔ ∀ᶠ (x : X) in 𝓝 x, p x
-- PartialHomeomorph.eventually_nhdsWithin
theorem eventually_nhdsWithin {x : X} (p : Y → Prop) {s : Set X}
    (hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (p : Y → Prop) {s : Set X} (hx : x ∈ e.source)
Goal: (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)
-- PartialHomeomorph.eventually_nhdsWithin'
theorem eventually_nhdsWithin' {x : X} (p : X → Prop) {s : Set X}
    (hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p (e.symm y)) ↔ ∀ᶠ x in 𝓝[s] x, p x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (p : X → Prop) {s : Set X} (hx : x ∈ e.source)
Goal: (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p (↑e.symm y)) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p x
-- PartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter
theorem preimage_eventuallyEq_target_inter_preimage_inter {e : PartialHomeomorph X Y} {s : Set X}
    {t : Set Z} {x : X} {f : X → Z} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source)
    (ht : t ∈ 𝓝 (f x)) :
    e.symm ⁻¹' s =ᶠ[𝓝 (e x)] (e.target ∩ e.symm ⁻¹' (s ∩ f ⁻¹' t) : Set Y) 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Z} {x : X} {f : X → Z} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source) (ht : t ∈ 𝓝 (f x))
Goal: ↑e.symm ⁻¹' s =ᶠ[𝓝 (↑e x)] e.target ∩ ↑e.symm ⁻¹' (s ∩ f ⁻¹' t)
-- PartialHomeomorph.isOpen_inter_preimage
theorem isOpen_inter_preimage {s : Set Y} (hs : IsOpen s) : IsOpen (e.source ∩ e ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set Y} (hs : IsOpen s)
Goal: IsOpen (e.source ∩ ↑e ⁻¹' s)
-- PartialHomeomorph.isOpen_inter_preimage_symm
theorem isOpen_inter_preimage_symm {s : Set X} (hs : IsOpen s) : IsOpen (e.target ∩ e.symm ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} (hs : IsOpen s)
Goal: IsOpen (e.target ∩ ↑e.symm ⁻¹' s)
-- PartialHomeomorph.isOpen_image_of_subset_source
lemma isOpen_image_of_subset_source {s : Set X} (hs : IsOpen s) (hse : s ⊆ e.source) :
    IsOpen (e '' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} (hs : IsOpen s) (hse : s ⊆ e.source)
Goal: IsOpen (↑e '' s)
-- PartialHomeomorph.isOpen_image_source_inter
theorem isOpen_image_source_inter {s : Set X} (hs : IsOpen s) :
    IsOpen (e '' (e.source ∩ s)) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} (hs : IsOpen s)
Goal: IsOpen (↑e '' (e.source ∩ s))
-- PartialHomeomorph.isOpen_image_symm_of_subset_target
lemma isOpen_image_symm_of_subset_target {t : Set Y} (ht : IsOpen t) (hte : t ⊆ e.target) :
    IsOpen (e.symm '' t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {t : Set Y} (ht : IsOpen t) (hte : t ⊆ e.target)
Goal: IsOpen (↑e.symm '' t)
-- PartialHomeomorph.isOpen_symm_image_iff_of_subset_target
lemma isOpen_symm_image_iff_of_subset_target {t : Set Y} (hs : t ⊆ e.target) :
    IsOpen (e.symm '' t) ↔ IsOpen t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {t : Set Y} (hs : t ⊆ e.target)
Goal: IsOpen (↑e.symm '' t) ↔ IsOpen t
-- PartialHomeomorph.isOpen_image_iff_of_subset_source
theorem isOpen_image_iff_of_subset_source {s : Set X} (hs : s ⊆ e.source) :
    IsOpen (e '' s) ↔ IsOpen s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} (hs : s ⊆ e.source)
Goal: IsOpen (↑e '' s) ↔ IsOpen s
-- PartialHomeomorph.IsImage.toPartialEquiv
theorem toPartialEquiv (h : e.IsImage s t) : e.toPartialEquiv.IsImage s t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: e.IsImage s t
-- PartialHomeomorph.IsImage.apply_mem_iff
theorem apply_mem_iff (h : e.IsImage s t) (hx : x ∈ e.source) : e x ∈ t ↔ x ∈ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {x : X} (h : e.IsImage s t) (hx : x ∈ e.source)
Goal: ↑e x ∈ t ↔ x ∈ s
-- PartialHomeomorph.IsImage.symm
protected theorem symm (h : e.IsImage s t) : e.symm.IsImage t s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: e.symm.IsImage t s
-- PartialHomeomorph.IsImage.symm_apply_mem_iff
theorem symm_apply_mem_iff (h : e.IsImage s t) (hy : y ∈ e.target) : e.symm y ∈ s ↔ y ∈ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {y : Y} (h : e.IsImage s t) (hy : y ∈ e.target)
Goal: ↑e.symm y ∈ s ↔ y ∈ t
-- PartialHomeomorph.IsImage.symm_iff
@[simp]
theorem symm_iff : e.symm.IsImage t s ↔ e.IsImage s t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y}
Goal: e.symm.IsImage t s ↔ e.IsImage s t
-- PartialHomeomorph.IsImage.mapsTo
protected theorem mapsTo (h : e.IsImage s t) : MapsTo e (e.source ∩ s) (e.target ∩ t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: Set.MapsTo (↑e) (e.source ∩ s) (e.target ∩ t)
-- PartialHomeomorph.IsImage.symm_mapsTo
theorem symm_mapsTo (h : e.IsImage s t) : MapsTo e.symm (e.target ∩ t) (e.source ∩ s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: Set.MapsTo (↑e.symm) (e.target ∩ t) (e.source ∩ s)
-- PartialHomeomorph.IsImage.image_eq
theorem image_eq (h : e.IsImage s t) : e '' (e.source ∩ s) = e.target ∩ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: ↑e '' (e.source ∩ s) = e.target ∩ t
-- PartialHomeomorph.IsImage.symm_image_eq
theorem symm_image_eq (h : e.IsImage s t) : e.symm '' (e.target ∩ t) = e.source ∩ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: ↑e.symm '' (e.target ∩ t) = e.source ∩ s
-- PartialHomeomorph.IsImage.iff_preimage_eq
theorem iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y}
Goal: e.IsImage s t ↔ e.source ∩ ↑e ⁻¹' t = e.source ∩ s
-- PartialHomeomorph.IsImage.iff_symm_preimage_eq
theorem iff_symm_preimage_eq : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' s = e.target ∩ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y}
Goal: e.IsImage s t ↔ e.target ∩ ↑e.symm ⁻¹' s = e.target ∩ t
-- PartialHomeomorph.IsImage.iff_symm_preimage_eq'
theorem iff_symm_preimage_eq' :
    e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y}
Goal: e.IsImage s t ↔ e.target ∩ ↑e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t
-- PartialHomeomorph.IsImage.iff_preimage_eq'
theorem iff_preimage_eq' : e.IsImage s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y}
Goal: e.IsImage s t ↔ e.source ∩ ↑e ⁻¹' (e.target ∩ t) = e.source ∩ s
-- PartialHomeomorph.IsImage.of_image_eq
theorem of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.IsImage s t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : ↑e '' (e.source ∩ s) = e.target ∩ t)
Goal: e.IsImage s t
-- PartialHomeomorph.IsImage.of_symm_image_eq
theorem of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.IsImage s t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : ↑e.symm '' (e.target ∩ t) = e.source ∩ s)
Goal: e.IsImage s t
-- PartialHomeomorph.IsImage.compl
protected theorem compl (h : e.IsImage s t) : e.IsImage sᶜ tᶜ 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: e.IsImage sᶜ tᶜ
-- PartialHomeomorph.IsImage.inter
protected theorem inter {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
    e.IsImage (s ∩ s') (t ∩ t') 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {s' : Set X} {t' : Set Y} (h : e.IsImage s t) (h' : e.IsImage s' t')
Goal: e.IsImage (s ∩ s') (t ∩ t')
-- PartialHomeomorph.IsImage.union
protected theorem union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
    e.IsImage (s ∪ s') (t ∪ t') 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {s' : Set X} {t' : Set Y} (h : e.IsImage s t) (h' : e.IsImage s' t')
Goal: e.IsImage (s ∪ s') (t ∪ t')
-- PartialHomeomorph.IsImage.diff
protected theorem diff {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') :
    e.IsImage (s \ s') (t \ t') 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {s' : Set X} {t' : Set Y} (h : e.IsImage s t) (h' : e.IsImage s' t')
Goal: e.IsImage (s \ s') (t \ t')
-- PartialHomeomorph.IsImage.leftInvOn_piecewise
theorem leftInvOn_piecewise {e' : PartialHomeomorph X Y} [∀ i, Decidable (i ∈ s)]
    [∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :
    LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {e' : PartialHomeomorph X Y} [(i : X) → Decidable (i ∈ s)] [(i : Y) → Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t)
Goal: Set.LeftInvOn (t.piecewise ↑e.symm ↑e'.symm) (s.piecewise ↑e ↑e') (s.ite e.source e'.source)
-- PartialHomeomorph.IsImage.inter_eq_of_inter_eq_of_eqOn
theorem inter_eq_of_inter_eq_of_eqOn {e' : PartialHomeomorph X Y} (h : e.IsImage s t)
    (h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :
    e.target ∩ t = e'.target ∩ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {e' : PartialHomeomorph X Y} (h : e.IsImage s t) (h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ s))
Goal: e.target ∩ t = e'.target ∩ t
-- PartialHomeomorph.IsImage.symm_eqOn_of_inter_eq_of_eqOn
theorem symm_eqOn_of_inter_eq_of_eqOn {e' : PartialHomeomorph X Y} (h : e.IsImage s t)
    (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) :
    EqOn e.symm e'.symm (e.target ∩ t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {e' : PartialHomeomorph X Y} (h : e.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ s))
Goal: Set.EqOn (↑e.symm) (↑e'.symm) (e.target ∩ t)
-- PartialHomeomorph.IsImage.map_nhdsWithin_eq
theorem map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} {x : X} (h : e.IsImage s t) (hx : x ∈ e.source)
Goal: Filter.map (↑e) (𝓝[s] x) = 𝓝[t] ↑e x
-- PartialHomeomorph.IsImage.closure
protected theorem closure (h : e.IsImage s t) : e.IsImage (closure s) (closure t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: e.IsImage (closure s) (closure t)
-- PartialHomeomorph.IsImage.interior
protected theorem interior (h : e.IsImage s t) : e.IsImage (interior s) (interior t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: e.IsImage (interior s) (interior t)
-- PartialHomeomorph.IsImage.frontier
protected theorem frontier (h : e.IsImage s t) : e.IsImage (frontier s) (frontier t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: e.IsImage (frontier s) (frontier t)
-- PartialHomeomorph.IsImage.isOpen_iff
theorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t)
Goal: IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t)
-- PartialHomeomorph.IsImage.restr_toPartialEquiv
@[simps toPartialEquiv]
def restr (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s)) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e : PartialHomeomorph X Y} {s : Set X} {t : Set Y} (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s))
Goal: (h.restr hs).toPartialEquiv = ⋯.restr
-- PartialHomeomorph.isImage_source_target
theorem isImage_source_target : e.IsImage e.source e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.IsImage e.source e.target
-- PartialHomeomorph.isImage_source_target_of_disjoint
theorem isImage_source_target_of_disjoint (e' : PartialHomeomorph X Y)
    (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) :
    e.IsImage e'.source e'.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e e' : PartialHomeomorph X Y) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target)
Goal: e.IsImage e'.source e'.target
-- PartialHomeomorph.preimage_interior
theorem preimage_interior (s : Set Y) :
    e.source ∩ e ⁻¹' interior s = e.source ∩ interior (e ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set Y)
Goal: e.source ∩ ↑e ⁻¹' interior s = e.source ∩ interior (↑e ⁻¹' s)
-- PartialHomeomorph.preimage_closure
theorem preimage_closure (s : Set Y) : e.source ∩ e ⁻¹' closure s = e.source ∩ closure (e ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set Y)
Goal: e.source ∩ ↑e ⁻¹' closure s = e.source ∩ closure (↑e ⁻¹' s)
-- PartialHomeomorph.preimage_frontier
theorem preimage_frontier (s : Set Y) :
    e.source ∩ e ⁻¹' frontier s = e.source ∩ frontier (e ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set Y)
Goal: e.source ∩ ↑e ⁻¹' frontier s = e.source ∩ frontier (↑e ⁻¹' s)
-- PartialHomeomorph.restrOpen_toPartialEquiv
@[simp, mfld_simps]
theorem restrOpen_toPartialEquiv (s : Set X) (hs : IsOpen s) :
    (e.restrOpen s hs).toPartialEquiv = e.toPartialEquiv.restr s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X) (hs : IsOpen s)
Goal: (e.restrOpen s hs).toPartialEquiv = e.restr s
-- PartialHomeomorph.restrOpen_source
theorem restrOpen_source (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).source = e.source ∩ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X) (hs : IsOpen s)
Goal: (e.restrOpen s hs).source = e.source ∩ s
-- PartialHomeomorph.restr_source
@[simps! (config := mfld_cfg) apply symm_apply, simps! (config := .lemmasOnly) source target]
protected def restr (s : Set X) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: (e.restr s).source = e.source ∩ interior s
-- PartialHomeomorph.restr_symm_apply
@[simps! (config := mfld_cfg) apply symm_apply, simps! (config := .lemmasOnly) source target]
protected def restr (s : Set X) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: ↑(e.restr s).symm = ↑e.symm
-- PartialHomeomorph.restr_apply
@[simps! (config := mfld_cfg) apply symm_apply, simps! (config := .lemmasOnly) source target]
protected def restr (s : Set X) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: ↑(e.restr s) = ↑e
-- PartialHomeomorph.restr_target
@[simps! (config := mfld_cfg) apply symm_apply, simps! (config := .lemmasOnly) source target]
protected def restr (s : Set X) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: (e.restr s).target = e.target ∩ ↑e.symm ⁻¹' interior s
-- PartialHomeomorph.restr_toPartialEquiv
@[simp, mfld_simps]
theorem restr_toPartialEquiv (s : Set X) :
    (e.restr s).toPartialEquiv = e.toPartialEquiv.restr (interior s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: (e.restr s).toPartialEquiv = e.restr (interior s)