ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsClosedMap.isClosed_range
theorem isClosed_range (hf : IsClosedMap f) : IsClosed (range f) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f)
Goal: IsClosed (Set.range f)
-- IsClosedMap.to_quotientMap
theorem to_quotientMap (hcl : IsClosedMap f) (hcont : Continuous f)
    (hsurj : Surjective f) : QuotientMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hcl : IsClosedMap f) (hcont : Continuous f) (hsurj : Function.Surjective f)
Goal: QuotientMap f
-- Inducing.isClosedMap
theorem Inducing.isClosedMap (hf : Inducing f) (h : IsClosed (range f)) : IsClosedMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) (h : IsClosed (Set.range f))
Goal: IsClosedMap f
-- isClosedMap_iff_closure_image
theorem isClosedMap_iff_closure_image :
    IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: IsClosedMap f ↔ ∀ (s : Set X), closure (f '' s) ⊆ f '' closure s
-- isClosedMap_iff_clusterPt
theorem isClosedMap_iff_clusterPt :
    IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: IsClosedMap f ↔ ∀ (s : Set X) (y : Y), MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s)
-- IsClosedMap.closure_image_eq_of_continuous
theorem IsClosedMap.closure_image_eq_of_continuous
    (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X) :
    closure (f '' s) = f '' closure s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X)
Goal: closure (f '' s) = f '' closure s
-- IsClosedMap.lift'_closure_map_eq
theorem IsClosedMap.lift'_closure_map_eq
    (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) :
    (map f F).lift' closure = map f (F.lift' closure) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X)
Goal: (Filter.map f F).lift' closure = Filter.map f (F.lift' closure)
-- IsClosedMap.mapClusterPt_iff_lift'_closure
theorem IsClosedMap.mapClusterPt_iff_lift'_closure
    {F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y} :
    MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y}
Goal: MapClusterPt y F f ↔ (F.lift' closure ⊓ 𝓟 (f ⁻¹' {y})).NeBot
-- OpenEmbedding.isOpenMap
theorem OpenEmbedding.isOpenMap (hf : OpenEmbedding f) : IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : OpenEmbedding f)
Goal: IsOpenMap f
-- OpenEmbedding.map_nhds_eq
theorem OpenEmbedding.map_nhds_eq (hf : OpenEmbedding f) (x : X) :
    map f (𝓝 x) = 𝓝 (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : OpenEmbedding f) (x : X)
Goal: Filter.map f (𝓝 x) = 𝓝 (f x)
-- OpenEmbedding.open_iff_image_open
theorem OpenEmbedding.open_iff_image_open (hf : OpenEmbedding f) {s : Set X} :
    IsOpen s ↔ IsOpen (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : OpenEmbedding f) {s : Set X}
Goal: IsOpen s ↔ IsOpen (f '' s)
-- OpenEmbedding.tendsto_nhds_iff
theorem OpenEmbedding.tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : OpenEmbedding g) :
    Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) 
Arguments: {Y : Type u_2} {Z : Type u_3} {ι : Type u_4} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y} (hg : OpenEmbedding g)
Goal: Filter.Tendsto f l (𝓝 y) ↔ Filter.Tendsto (g ∘ f) l (𝓝 (g y))
-- OpenEmbedding.tendsto_nhds_iff'
theorem OpenEmbedding.tendsto_nhds_iff' (hf : OpenEmbedding f) {l : Filter Z} {x : X} :
    Tendsto (g ∘ f) (𝓝 x) l ↔ Tendsto g (𝓝 (f x)) l 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] (hf : OpenEmbedding f) {l : Filter Z} {x : X}
Goal: Filter.Tendsto (g ∘ f) (𝓝 x) l ↔ Filter.Tendsto g (𝓝 (f x)) l
-- OpenEmbedding.continuousAt_iff
theorem OpenEmbedding.continuousAt_iff (hf : OpenEmbedding f) {x : X} :
    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : OpenEmbedding f) {x : X}
Goal: ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)
-- OpenEmbedding.continuous
theorem OpenEmbedding.continuous (hf : OpenEmbedding f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : OpenEmbedding f)
Goal: Continuous f
-- OpenEmbedding.open_iff_preimage_open
theorem OpenEmbedding.open_iff_preimage_open (hf : OpenEmbedding f) {s : Set Y}
    (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : OpenEmbedding f) {s : Set Y} (hs : s ⊆ Set.range f)
Goal: IsOpen s ↔ IsOpen (f ⁻¹' s)
-- openEmbedding_of_embedding_open
theorem openEmbedding_of_embedding_open (h₁ : Embedding f) (h₂ : IsOpenMap f) :
    OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Embedding f) (h₂ : IsOpenMap f)
Goal: OpenEmbedding f
-- Embedding.toOpenEmbedding_of_surjective
theorem _root_.Embedding.toOpenEmbedding_of_surjective (hf : Embedding f) (hsurj : f.Surjective) :
    OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Embedding f) (hsurj : Function.Surjective f)
Goal: OpenEmbedding f
-- openEmbedding_iff_embedding_open
theorem openEmbedding_iff_embedding_open :
    OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f
-- openEmbedding_of_continuous_injective_open
theorem openEmbedding_of_continuous_injective_open
    (h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Continuous f) (h₂ : Function.Injective f) (h₃ : IsOpenMap f)
Goal: OpenEmbedding f
-- openEmbedding_iff_continuous_injective_open
theorem openEmbedding_iff_continuous_injective_open :
    OpenEmbedding f ↔ Continuous f ∧ Injective f ∧ IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: OpenEmbedding f ↔ Continuous f ∧ Function.Injective f ∧ IsOpenMap f
-- openEmbedding_id
theorem openEmbedding_id : OpenEmbedding (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: OpenEmbedding id
-- OpenEmbedding.comp
protected theorem comp (hg : OpenEmbedding g)
    (hf : OpenEmbedding f) : OpenEmbedding (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : OpenEmbedding g) (hf : OpenEmbedding f)
Goal: OpenEmbedding (g ∘ f)
-- OpenEmbedding.isOpenMap_iff
theorem isOpenMap_iff (hg : OpenEmbedding g) :
    IsOpenMap f ↔ IsOpenMap (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : OpenEmbedding g)
Goal: IsOpenMap f ↔ IsOpenMap (g ∘ f)
-- OpenEmbedding.of_comp_iff
theorem of_comp_iff (f : X → Y) (hg : OpenEmbedding g) :
    OpenEmbedding (g ∘ f) ↔ OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → Y) (hg : OpenEmbedding g)
Goal: OpenEmbedding (g ∘ f) ↔ OpenEmbedding f
-- OpenEmbedding.of_comp
theorem of_comp (f : X → Y) (hg : OpenEmbedding g)
    (h : OpenEmbedding (g ∘ f)) : OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → Y) (hg : OpenEmbedding g) (h : OpenEmbedding (g ∘ f))
Goal: OpenEmbedding f
-- OpenEmbedding.of_isEmpty
theorem of_isEmpty [IsEmpty X] (f : X → Y) : OpenEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [IsEmpty X] (f : X → Y)
Goal: OpenEmbedding f
-- ClosedEmbedding.tendsto_nhds_iff
theorem tendsto_nhds_iff {g : ι → X} {l : Filter ι} {x : X} (hf : ClosedEmbedding f) :
    Tendsto g l (𝓝 x) ↔ Tendsto (f ∘ g) l (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} {ι : Type u_4} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {g : ι → X} {l : Filter ι} {x : X} (hf : ClosedEmbedding f)
Goal: Filter.Tendsto g l (𝓝 x) ↔ Filter.Tendsto (f ∘ g) l (𝓝 (f x))
-- ClosedEmbedding.continuous
theorem continuous (hf : ClosedEmbedding f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ClosedEmbedding f)
Goal: Continuous f
-- ClosedEmbedding.isClosedMap
theorem isClosedMap (hf : ClosedEmbedding f) : IsClosedMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ClosedEmbedding f)
Goal: IsClosedMap f
-- ClosedEmbedding.closed_iff_image_closed
theorem closed_iff_image_closed (hf : ClosedEmbedding f) {s : Set X} :
    IsClosed s ↔ IsClosed (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ClosedEmbedding f) {s : Set X}
Goal: IsClosed s ↔ IsClosed (f '' s)
-- ClosedEmbedding.closed_iff_preimage_closed
theorem closed_iff_preimage_closed (hf : ClosedEmbedding f) {s : Set Y}
    (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ClosedEmbedding f) {s : Set Y} (hs : s ⊆ Set.range f)
Goal: IsClosed s ↔ IsClosed (f ⁻¹' s)
-- closedEmbedding_of_embedding_closed
theorem _root_.closedEmbedding_of_embedding_closed (h₁ : Embedding f) (h₂ : IsClosedMap f) :
    ClosedEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Embedding f) (h₂ : IsClosedMap f)
Goal: ClosedEmbedding f
-- closedEmbedding_of_continuous_injective_closed
theorem _root_.closedEmbedding_of_continuous_injective_closed (h₁ : Continuous f) (h₂ : Injective f)
    (h₃ : IsClosedMap f) : ClosedEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Continuous f) (h₂ : Function.Injective f) (h₃ : IsClosedMap f)
Goal: ClosedEmbedding f
-- closedEmbedding_id
theorem _root_.closedEmbedding_id : ClosedEmbedding (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: ClosedEmbedding id
-- ClosedEmbedding.comp
theorem comp (hg : ClosedEmbedding g) (hf : ClosedEmbedding f) :
    ClosedEmbedding (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : ClosedEmbedding g) (hf : ClosedEmbedding f)
Goal: ClosedEmbedding (g ∘ f)
-- ClosedEmbedding.of_comp_iff
theorem of_comp_iff (hg : ClosedEmbedding g) :
    ClosedEmbedding (g ∘ f) ↔ ClosedEmbedding f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : ClosedEmbedding g)
Goal: ClosedEmbedding (g ∘ f) ↔ ClosedEmbedding f
-- ClosedEmbedding.closure_image_eq
theorem closure_image_eq (hf : ClosedEmbedding f) (s : Set X) :
    closure (f '' s) = f '' closure s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ClosedEmbedding f) (s : Set X)
Goal: closure (f '' s) = f '' closure s
-- isProperMap_iff_clusterPt
@[mk_iff isProperMap_iff_clusterPt, fun_prop]
structure IsProperMap (f : X → Y) extends Continuous f : Prop where
    clusterPt_of_mapClusterPt :
    ∀ ⦃ℱ : Filter X⦄, ∀ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y)
Goal: IsProperMap f ↔ Continuous f ∧ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
-- IsProperMap.clusterPt_of_mapClusterPt
@[mk_iff isProperMap_iff_clusterPt, fun_prop]
structure IsProperMap (f : X → Y) extends Continuous f : Prop where
    clusterPt_of_mapClusterPt :
    ∀ ⦃ℱ : Filter X⦄, ∀ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (self : IsProperMap f) ⦃ℱ : Filter X⦄ ⦃y : Y⦄
Goal: MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ
-- IsProperMap.continuous
@[fun_prop]
lemma IsProperMap.continuous (h : IsProperMap f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : IsProperMap f)
Goal: Continuous f
-- IsProperMap.isClosedMap
lemma IsProperMap.isClosedMap (h : IsProperMap f) : IsClosedMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : IsProperMap f)
Goal: IsClosedMap f
-- isProperMap_iff_ultrafilter
lemma isProperMap_iff_ultrafilter : IsProperMap f ↔ Continuous f ∧
    ∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsProperMap f ↔ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Filter.Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
-- isProperMap_iff_ultrafilter_of_t2
lemma isProperMap_iff_ultrafilter_of_t2 [T2Space Y] : IsProperMap f ↔ Continuous f ∧
    ∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, 𝒰.1 ≤ 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y]
Goal: IsProperMap f ↔ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Filter.Tendsto f (↑𝒰) (𝓝 y) → ∃ x, ↑𝒰 ≤ 𝓝 x
-- IsProperMap.ultrafilter_le_nhds_of_tendsto
lemma IsProperMap.ultrafilter_le_nhds_of_tendsto (h : IsProperMap f) ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄
    (hy : Tendsto f 𝒰 (𝓝 y)) : ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : IsProperMap f) ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄ (hy : Filter.Tendsto f (↑𝒰) (𝓝 y))
Goal: ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
-- IsProperMap.comp
lemma IsProperMap.comp (hf : IsProperMap f) (hg : IsProperMap g) :
    IsProperMap (g ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} (hf : IsProperMap f) (hg : IsProperMap g)
Goal: IsProperMap (g ∘ f)
-- isProperMap_of_comp_of_surj
lemma isProperMap_of_comp_of_surj (hf : Continuous f)
    (hg : Continuous g) (hgf : IsProperMap (g ∘ f)) (f_surj : f.Surjective) : IsProperMap g 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} (hf : Continuous f) (hg : Continuous g) (hgf : IsProperMap (g ∘ f)) (f_surj : Function.Surjective f)
Goal: IsProperMap g
-- isProperMap_of_comp_of_inj
lemma isProperMap_of_comp_of_inj {f : X → Y} {g : Y → Z} (hf : Continuous f) (hg : Continuous g)
    (hgf : IsProperMap (g ∘ f)) (g_inj : g.Injective) : IsProperMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} (hf : Continuous f) (hg : Continuous g) (hgf : IsProperMap (g ∘ f)) (g_inj : Function.Injective g)
Goal: IsProperMap f
-- isProperMap_of_comp_of_t2
lemma isProperMap_of_comp_of_t2 [T2Space Y] (hf : Continuous f) (hg : Continuous g)
    (hgf : IsProperMap (g ∘ f)) : IsProperMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} {g : Y → Z} [T2Space Y] (hf : Continuous f) (hg : Continuous g) (hgf : IsProperMap (g ∘ f))
Goal: IsProperMap f
-- IsProperMap.prod_map
lemma IsProperMap.prod_map {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :
    IsProperMap (Prod.map f g) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {W : Type u_4} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : X → Y} {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g)
Goal: IsProperMap (Prod.map f g)
-- IsProperMap.pi_map
lemma IsProperMap.pi_map {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
    [∀ i, TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (h : ∀ i, IsProperMap (f i)) :
    IsProperMap (fun (x : ∀ i, X i) i ↦ f i (x i)) 
Arguments: {ι : Type u_5} {X : ι → Type u_6} {Y : ι → Type u_7} [(i : ι) → TopologicalSpace (X i)] [(i : ι) → TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (h : ∀ (i : ι), IsProperMap (f i))
Goal: IsProperMap fun x i => f i (x i)
-- IsProperMap.isCompact_preimage
lemma IsProperMap.isCompact_preimage (h : IsProperMap f) {K : Set Y} (hK : IsCompact K) :
    IsCompact (f ⁻¹' K) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (h : IsProperMap f) {K : Set Y} (hK : IsCompact K)
Goal: IsCompact (f ⁻¹' K)
-- isProperMap_iff_isClosedMap_and_compact_fibers
theorem isProperMap_iff_isClosedMap_and_compact_fibers :
    IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ ∀ y, IsCompact (f ⁻¹' {y}) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y}
Goal: IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ ∀ (y : Y), IsCompact (f ⁻¹' {y})
-- isProperMap_iff_isClosedMap_of_inj
lemma isProperMap_iff_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) :
    IsProperMap f ↔ IsClosedMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (f_cont : Continuous f) (f_inj : Function.Injective f)
Goal: IsProperMap f ↔ IsClosedMap f
-- isProperMap_of_isClosedMap_of_inj
lemma isProperMap_of_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective)
    (f_closed : IsClosedMap f) : IsProperMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (f_cont : Continuous f) (f_inj : Function.Injective f) (f_closed : IsClosedMap f)
Goal: IsProperMap f
-- Homeomorph.isProperMap
@[simp] lemma Homeomorph.isProperMap (e : X ≃ₜ Y) : IsProperMap e 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: IsProperMap ⇑e
-- isProperMap_id
@[simp] lemma isProperMap_id : IsProperMap (id : X → X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsProperMap id
-- isProperMap_of_closedEmbedding
lemma isProperMap_of_closedEmbedding (hf : ClosedEmbedding f) : IsProperMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : ClosedEmbedding f)
Goal: IsProperMap f
-- isProperMap_subtype_val_of_closed
lemma isProperMap_subtype_val_of_closed {U : Set X} (hU : IsClosed U) : IsProperMap ((↑) : U → X) 
Arguments: {X : Type u_1} [TopologicalSpace X] {U : Set X} (hU : IsClosed U)
Goal: IsProperMap Subtype.val
-- isProperMap_restr_of_proper_of_closed
lemma isProperMap_restr_of_proper_of_closed {U : Set X} (hf : IsProperMap f) (hU : IsClosed U) :
    IsProperMap (fun x : U ↦ f x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {U : Set X} (hf : IsProperMap f) (hU : IsClosed U)
Goal: IsProperMap fun x => f ↑x
-- IsProperMap.isClosed_range
lemma IsProperMap.isClosed_range (hf : IsProperMap f) : IsClosed (range f) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsProperMap f)
Goal: IsClosed (Set.range f)
-- isProperMap_iff_isClosedMap_and_tendsto_cofinite
lemma isProperMap_iff_isClosedMap_and_tendsto_cofinite [T1Space Y] :
    IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ Tendsto f (cocompact X) cofinite 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T1Space Y]
Goal: IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ Filter.Tendsto f (Filter.cocompact X) Filter.cofinite
-- Continuous.isProperMap
theorem Continuous.isProperMap [CompactSpace X] [T2Space Y] (hf : Continuous f) : IsProperMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [CompactSpace X] [T2Space Y] (hf : Continuous f)
Goal: IsProperMap f
-- isProperMap_iff_isCompact_preimage
theorem isProperMap_iff_isCompact_preimage [T2Space Y]
    (compactlyGenerated : ∀ s : Set Y, IsClosed s ↔ ∀ ⦃K⦄, IsCompact K → IsClosed (s ∩ K)) :
    IsProperMap f ↔ Continuous f ∧ ∀ ⦃K⦄, IsCompact K → IsCompact (f ⁻¹' K) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y] (compactlyGenerated : ∀ (s : Set Y), IsClosed s ↔ ∀ ⦃K : Set Y⦄, IsCompact K → IsClosed (s ∩ K))
Goal: IsProperMap f ↔ Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)
-- compactlyGenerated_of_weaklyLocallyCompactSpace
theorem compactlyGenerated_of_weaklyLocallyCompactSpace [T2Space X] [WeaklyLocallyCompactSpace X]
    {s : Set X} : IsClosed s ↔ ∀ ⦃K⦄, IsCompact K → IsClosed (s ∩ K) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [WeaklyLocallyCompactSpace X] {s : Set X}
Goal: IsClosed s ↔ ∀ ⦃K : Set X⦄, IsCompact K → IsClosed (s ∩ K)
-- WeaklyLocallyCompactSpace.isProperMap_iff_isCompact_preimage
theorem WeaklyLocallyCompactSpace.isProperMap_iff_isCompact_preimage [T2Space Y]
    [WeaklyLocallyCompactSpace Y] :
    IsProperMap f ↔ Continuous f ∧ ∀ ⦃K⦄, IsCompact K → IsCompact (f ⁻¹' K) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y] [WeaklyLocallyCompactSpace Y]
Goal: IsProperMap f ↔ Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)
-- compactlyGenerated_of_sequentialSpace
theorem compactlyGenerated_of_sequentialSpace [T2Space X] [SequentialSpace X] {s : Set X} :
    IsClosed s ↔ ∀ ⦃K⦄, IsCompact K → IsClosed (s ∩ K) 
Arguments: {X : Type u_1} [TopologicalSpace X] [T2Space X] [SequentialSpace X] {s : Set X}
Goal: IsClosed s ↔ ∀ ⦃K : Set X⦄, IsCompact K → IsClosed (s ∩ K)
-- SequentialSpace.isProperMap_iff_isCompact_preimage
theorem SequentialSpace.isProperMap_iff_isCompact_preimage [T2Space Y] [SequentialSpace Y] :
    IsProperMap f ↔ Continuous f ∧ ∀ ⦃K⦄, IsCompact K → IsCompact (f ⁻¹' K) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y] [SequentialSpace Y]
Goal: IsProperMap f ↔ Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)
-- isProperMap_iff_tendsto_cocompact
lemma isProperMap_iff_tendsto_cocompact [T2Space Y]
    (compactlyGenerated : ∀ s : Set Y, IsClosed s ↔ ∀ ⦃K⦄, IsCompact K → IsClosed (s ∩ K)) :
    IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y] (compactlyGenerated : ∀ (s : Set Y), IsClosed s ↔ ∀ ⦃K : Set Y⦄, IsCompact K → IsClosed (s ∩ K))
Goal: IsProperMap f ↔ Continuous f ∧ Filter.Tendsto f (Filter.cocompact X) (Filter.cocompact Y)
-- WeaklyLocallyCompactSpace.isProperMap_iff_tendsto_cocompact
lemma WeaklyLocallyCompactSpace.isProperMap_iff_tendsto_cocompact [T2Space Y]
    [WeaklyLocallyCompactSpace Y] :
    IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [T2Space Y] [WeaklyLocallyCompactSpace Y]
Goal: IsProperMap f ↔ Continuous f ∧ Filter.Tendsto f (Filter.cocompact X) (Filter.cocompact Y)