ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isConnected_connectedComponent
theorem isConnected_connectedComponent {x : α} : IsConnected (connectedComponent x) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsConnected (connectedComponent x)
-- isConnected_connectedComponentIn_iff
theorem isConnected_connectedComponentIn_iff {x : α} {F : Set α} :
    IsConnected (connectedComponentIn F x) ↔ x ∈ F 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {F : Set α}
Goal: IsConnected (connectedComponentIn F x) ↔ x ∈ F
-- IsPreconnected.subset_connectedComponent
theorem IsPreconnected.subset_connectedComponent {x : α} {s : Set α} (H1 : IsPreconnected s)
    (H2 : x ∈ s) : s ⊆ connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {s : Set α} (H1 : IsPreconnected s) (H2 : x ∈ s)
Goal: s ⊆ connectedComponent x
-- IsPreconnected.subset_connectedComponentIn
theorem IsPreconnected.subset_connectedComponentIn {x : α} {F : Set α} (hs : IsPreconnected s)
    (hxs : x ∈ s) (hsF : s ⊆ F) : s ⊆ connectedComponentIn F x 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} {x : α} {F : Set α} (hs : IsPreconnected s) (hxs : x ∈ s) (hsF : s ⊆ F)
Goal: s ⊆ connectedComponentIn F x
-- IsConnected.subset_connectedComponent
theorem IsConnected.subset_connectedComponent {x : α} {s : Set α} (H1 : IsConnected s)
    (H2 : x ∈ s) : s ⊆ connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {s : Set α} (H1 : IsConnected s) (H2 : x ∈ s)
Goal: s ⊆ connectedComponent x
-- IsPreconnected.connectedComponentIn
theorem IsPreconnected.connectedComponentIn {x : α} {F : Set α} (h : IsPreconnected F)
    (hx : x ∈ F) : connectedComponentIn F x = F 
Arguments: {α : Type u} [TopologicalSpace α] {x : α} {F : Set α} (h : IsPreconnected F) (hx : x ∈ F)
Goal: connectedComponentIn F x = F
-- connectedComponent_eq
theorem connectedComponent_eq {x y : α} (h : y ∈ connectedComponent x) :
    connectedComponent x = connectedComponent y 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α} (h : y ∈ connectedComponent x)
Goal: connectedComponent x = connectedComponent y
-- connectedComponent_eq_iff_mem
theorem connectedComponent_eq_iff_mem {x y : α} :
    connectedComponent x = connectedComponent y ↔ x ∈ connectedComponent y 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α}
Goal: connectedComponent x = connectedComponent y ↔ x ∈ connectedComponent y
-- connectedComponentIn_eq
theorem connectedComponentIn_eq {x y : α} {F : Set α} (h : y ∈ connectedComponentIn F x) :
    connectedComponentIn F x = connectedComponentIn F y 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α} {F : Set α} (h : y ∈ connectedComponentIn F x)
Goal: connectedComponentIn F x = connectedComponentIn F y
-- connectedComponentIn_univ
theorem connectedComponentIn_univ (x : α) : connectedComponentIn univ x = connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] (x : α)
Goal: connectedComponentIn Set.univ x = connectedComponent x
-- connectedComponent_disjoint
theorem connectedComponent_disjoint {x y : α} (h : connectedComponent x ≠ connectedComponent y) :
    Disjoint (connectedComponent x) (connectedComponent y) 
Arguments: {α : Type u} [TopologicalSpace α] {x y : α} (h : connectedComponent x ≠ connectedComponent y)
Goal: Disjoint (connectedComponent x) (connectedComponent y)
-- isClosed_connectedComponent
theorem isClosed_connectedComponent {x : α} : IsClosed (connectedComponent x) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsClosed (connectedComponent x)
-- Continuous.image_connectedComponent_subset
theorem Continuous.image_connectedComponent_subset [TopologicalSpace β] {f : α → β}
    (h : Continuous f) (a : α) : f '' connectedComponent a ⊆ connectedComponent (f a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (h : Continuous f) (a : α)
Goal: f '' connectedComponent a ⊆ connectedComponent (f a)
-- Continuous.image_connectedComponentIn_subset
theorem Continuous.image_connectedComponentIn_subset [TopologicalSpace β] {f : α → β} {s : Set α}
    {a : α} (hf : Continuous f) (hx : a ∈ s) :
    f '' connectedComponentIn s a ⊆ connectedComponentIn (f '' s) (f a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {a : α} (hf : Continuous f) (hx : a ∈ s)
Goal: f '' connectedComponentIn s a ⊆ connectedComponentIn (f '' s) (f a)
-- Continuous.mapsTo_connectedComponent
theorem Continuous.mapsTo_connectedComponent [TopologicalSpace β] {f : α → β} (h : Continuous f)
    (a : α) : MapsTo f (connectedComponent a) (connectedComponent (f a)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (h : Continuous f) (a : α)
Goal: Set.MapsTo f (connectedComponent a) (connectedComponent (f a))
-- Continuous.mapsTo_connectedComponentIn
theorem Continuous.mapsTo_connectedComponentIn [TopologicalSpace β] {f : α → β} {s : Set α}
    (h : Continuous f) {a : α} (hx : a ∈ s) :
    MapsTo f (connectedComponentIn s a) (connectedComponentIn (f '' s) (f a)) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} (h : Continuous f) {a : α} (hx : a ∈ s)
Goal: Set.MapsTo f (connectedComponentIn s a) (connectedComponentIn (f '' s) (f a))
-- irreducibleComponent_subset_connectedComponent
theorem irreducibleComponent_subset_connectedComponent {x : α} :
    irreducibleComponent x ⊆ connectedComponent x 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: irreducibleComponent x ⊆ connectedComponent x
-- connectedComponentIn_mono
@[mono]
theorem connectedComponentIn_mono (x : α) {F G : Set α} (h : F ⊆ G) :
    connectedComponentIn F x ⊆ connectedComponentIn G x 
Arguments: {α : Type u} [TopologicalSpace α] (x : α) {F G : Set α} (h : F ⊆ G)
Goal: connectedComponentIn F x ⊆ connectedComponentIn G x
-- PreconnectedSpace.isPreconnected_univ
class PreconnectedSpace (α : Type u) [TopologicalSpace α] : Prop where
    isPreconnected_univ : IsPreconnected (univ : Set α)
Arguments: {α : Type u} [TopologicalSpace α] [self : PreconnectedSpace α]
Goal: IsPreconnected Set.univ
-- ConnectedSpace.toNonempty
class ConnectedSpace (α : Type u) [TopologicalSpace α] extends PreconnectedSpace α : Prop where
    toNonempty : Nonempty α
Arguments: {α : Type u} [TopologicalSpace α] [self : ConnectedSpace α]
Goal: Nonempty α
-- isConnected_univ
theorem isConnected_univ [ConnectedSpace α] : IsConnected (univ : Set α) 
Arguments: {α : Type u} [TopologicalSpace α] [ConnectedSpace α]
Goal: IsConnected Set.univ
-- preconnectedSpace_iff_univ
lemma preconnectedSpace_iff_univ : PreconnectedSpace α ↔ IsPreconnected (univ : Set α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: PreconnectedSpace α ↔ IsPreconnected Set.univ
-- connectedSpace_iff_univ
lemma connectedSpace_iff_univ : ConnectedSpace α ↔ IsConnected (univ : Set α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: ConnectedSpace α ↔ IsConnected Set.univ
-- isPreconnected_range
theorem isPreconnected_range [TopologicalSpace β] [PreconnectedSpace α] {f : α → β}
    (h : Continuous f) : IsPreconnected (range f) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [PreconnectedSpace α] {f : α → β} (h : Continuous f)
Goal: IsPreconnected (Set.range f)
-- isConnected_range
theorem isConnected_range [TopologicalSpace β] [ConnectedSpace α] {f : α → β} (h : Continuous f) :
    IsConnected (range f) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [ConnectedSpace α] {f : α → β} (h : Continuous f)
Goal: IsConnected (Set.range f)
-- Function.Surjective.connectedSpace
theorem Function.Surjective.connectedSpace [ConnectedSpace α] [TopologicalSpace β]
    {f : α → β} (hf : Surjective f) (hf' : Continuous f) : ConnectedSpace β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [ConnectedSpace α] [TopologicalSpace β] {f : α → β} (hf : Function.Surjective f) (hf' : Continuous f)
Goal: ConnectedSpace β
-- DenseRange.preconnectedSpace
theorem DenseRange.preconnectedSpace [TopologicalSpace β] [PreconnectedSpace α] {f : α → β}
    (hf : DenseRange f) (hc : Continuous f) : PreconnectedSpace β 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [PreconnectedSpace α] {f : α → β} (hf : DenseRange f) (hc : Continuous f)
Goal: PreconnectedSpace β
-- connectedSpace_iff_connectedComponent
theorem connectedSpace_iff_connectedComponent :
    ConnectedSpace α ↔ ∃ x : α, connectedComponent x = univ 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: ConnectedSpace α ↔ ∃ x, connectedComponent x = Set.univ
-- preconnectedSpace_iff_connectedComponent
theorem preconnectedSpace_iff_connectedComponent :
    PreconnectedSpace α ↔ ∀ x : α, connectedComponent x = univ 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: PreconnectedSpace α ↔ ∀ (x : α), connectedComponent x = Set.univ
-- PreconnectedSpace.connectedComponent_eq_univ
@[simp]
theorem PreconnectedSpace.connectedComponent_eq_univ {X : Type*} [TopologicalSpace X]
    [h : PreconnectedSpace X] (x : X) : connectedComponent x = univ 
Arguments: {X : Type u_3} [TopologicalSpace X] [h : PreconnectedSpace X] (x : X)
Goal: connectedComponent x = Set.univ
-- Subtype.preconnectedSpace
theorem Subtype.preconnectedSpace {s : Set α} (h : IsPreconnected s) : PreconnectedSpace s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (h : IsPreconnected s)
Goal: PreconnectedSpace ↑s
-- Subtype.connectedSpace
theorem Subtype.connectedSpace {s : Set α} (h : IsConnected s) : ConnectedSpace s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (h : IsConnected s)
Goal: ConnectedSpace ↑s
-- isPreconnected_iff_preconnectedSpace
theorem isPreconnected_iff_preconnectedSpace {s : Set α} : IsPreconnected s ↔ PreconnectedSpace s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: IsPreconnected s ↔ PreconnectedSpace ↑s
-- isConnected_iff_connectedSpace
theorem isConnected_iff_connectedSpace {s : Set α} : IsConnected s ↔ ConnectedSpace s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: IsConnected s ↔ ConnectedSpace ↑s
-- uniqueMDiffWithinAt_iff_uniqueDiffWithinAt
theorem uniqueMDiffWithinAt_iff_uniqueDiffWithinAt :
    UniqueMDiffWithinAt 𝓘(𝕜, E) s x ↔ UniqueDiffWithinAt 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {x : E}
Goal: UniqueMDiffWithinAt 𝓘(𝕜, E) s x ↔ UniqueDiffWithinAt 𝕜 s x
-- uniqueMDiffOn_iff_uniqueDiffOn
theorem uniqueMDiffOn_iff_uniqueDiffOn : UniqueMDiffOn 𝓘(𝕜, E) s ↔ UniqueDiffOn 𝕜 s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: UniqueMDiffOn 𝓘(𝕜, E) s ↔ UniqueDiffOn 𝕜 s
-- writtenInExtChartAt_model_space
theorem writtenInExtChartAt_model_space : writtenInExtChartAt 𝓘(𝕜, E) 𝓘(𝕜, E') x f = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {x : E}
Goal: writtenInExtChartAt 𝓘(𝕜, E) 𝓘(𝕜, E') x f = f
-- hasMFDerivWithinAt_iff_hasFDerivWithinAt
theorem hasMFDerivWithinAt_iff_hasFDerivWithinAt {f'} :
    HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x f' ↔ HasFDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {s : Set E} {x : E} {f' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)}
Goal: HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x f' ↔ HasFDerivWithinAt f f' s x
-- hasMFDerivAt_iff_hasFDerivAt
theorem hasMFDerivAt_iff_hasFDerivAt {f'} :
    HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x f' ↔ HasFDerivAt f f' x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {x : E} {f' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)}
Goal: HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x f' ↔ HasFDerivAt f f' x
-- mdifferentiableWithinAt_iff_differentiableWithinAt
theorem mdifferentiableWithinAt_iff_differentiableWithinAt :
    MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x ↔ DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {s : Set E} {x : E}
Goal: MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x ↔ DifferentiableWithinAt 𝕜 f s x
-- mdifferentiableAt_iff_differentiableAt
theorem mdifferentiableAt_iff_differentiableAt :
    MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x ↔ DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {x : E}
Goal: MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x ↔ DifferentiableAt 𝕜 f x
-- mdifferentiableOn_iff_differentiableOn
theorem mdifferentiableOn_iff_differentiableOn :
    MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') f s ↔ DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {s : Set E}
Goal: MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') f s ↔ DifferentiableOn 𝕜 f s
-- mdifferentiable_iff_differentiable
theorem mdifferentiable_iff_differentiable :
    MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') f ↔ Differentiable 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
Goal: MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') f ↔ Differentiable 𝕜 f
-- mfderivWithin_eq_fderivWithin
@[simp]
theorem mfderivWithin_eq_fderivWithin :
    mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') f s x = fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {s : Set E} {x : E}
Goal: mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') f s x = fderivWithin 𝕜 f s x
-- mfderiv_eq_fderiv
@[simp]
theorem mfderiv_eq_fderiv : mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') f x = fderiv 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'} {x : E}
Goal: mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') f x = fderiv 𝕜 f x
-- UniqueMDiffWithinAt.image_denseRange
theorem UniqueMDiffWithinAt.image_denseRange (hs : UniqueMDiffWithinAt I s x)
    {f : M → M'} {f' : E →L[𝕜] E'} (hf : HasMFDerivWithinAt I I' f s x f')
    (hd : DenseRange f') : UniqueMDiffWithinAt I' (f '' s) (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x) {f : M → M'} {f' : E →L[𝕜] E'} (hf : HasMFDerivWithinAt I I' f s x f') (hd : DenseRange ⇑f')
Goal: UniqueMDiffWithinAt I' (f '' s) (f x)
-- UniqueMDiffOn.image_denseRange'
theorem UniqueMDiffOn.image_denseRange' (hs : UniqueMDiffOn I s) {f : M → M'}
    {f' : M → E →L[𝕜] E'} (hf : ∀ x ∈ s, HasMFDerivWithinAt I I' f s x (f' x))
    (hd : ∀ x ∈ s, DenseRange (f' x)) :
    UniqueMDiffOn I' (f '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set M} (hs : UniqueMDiffOn I s) {f : M → M'} {f' : M → E →L[𝕜] E'} (hf : ∀ x ∈ s, HasMFDerivWithinAt I I' f s x (f' x)) (hd : ∀ x ∈ s, DenseRange ⇑(f' x))
Goal: UniqueMDiffOn I' (f '' s)
-- UniqueMDiffOn.image_denseRange
theorem UniqueMDiffOn.image_denseRange (hs : UniqueMDiffOn I s) {f : M → M'}
    (hf : MDifferentiableOn I I' f s) (hd : ∀ x ∈ s, DenseRange (mfderivWithin I I' f s x)) :
    UniqueMDiffOn I' (f '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set M} (hs : UniqueMDiffOn I s) {f : M → M'} (hf : MDifferentiableOn I I' f s) (hd : ∀ x ∈ s, DenseRange ⇑(mfderivWithin I I' f s x))
Goal: UniqueMDiffOn I' (f '' s)
-- UniqueMDiffWithinAt.preimage_partialHomeomorph
protected theorem UniqueMDiffWithinAt.preimage_partialHomeomorph (hs : UniqueMDiffWithinAt I s x)
    {e : PartialHomeomorph M M'} (he : e.MDifferentiable I I') (hx : x ∈ e.source) :
    UniqueMDiffWithinAt I' (e.target ∩ e.symm ⁻¹' s) (e x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x) {e : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) (hx : x ∈ e.source)
Goal: UniqueMDiffWithinAt I' (e.target ∩ ↑e.symm ⁻¹' s) (↑e x)
-- UniqueMDiffOn.uniqueMDiffOn_preimage
theorem UniqueMDiffOn.uniqueMDiffOn_preimage (hs : UniqueMDiffOn I s) {e : PartialHomeomorph M M'}
    (he : e.MDifferentiable I I') : UniqueMDiffOn I' (e.target ∩ e.symm ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [SmoothManifoldWithCorners I' M'] {s : Set M} (hs : UniqueMDiffOn I s) {e : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e)
Goal: UniqueMDiffOn I' (e.target ∩ ↑e.symm ⁻¹' s)
-- UniqueMDiffOn.uniqueDiffOn_target_inter
theorem UniqueMDiffOn.uniqueDiffOn_target_inter (hs : UniqueMDiffOn I s) (x : M) :
    UniqueDiffOn 𝕜 ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {s : Set M} (hs : UniqueMDiffOn I s) (x : M)
Goal: UniqueDiffOn 𝕜 ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)
-- UniqueMDiffOn.uniqueDiffOn_inter_preimage
theorem UniqueMDiffOn.uniqueDiffOn_inter_preimage (hs : UniqueMDiffOn I s) (x : M) (y : M')
    {f : M → M'} (hf : ContinuousOn f s) :
    UniqueDiffOn 𝕜
      ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {s : Set M} (hs : UniqueMDiffOn I s) (x : M) (y : M') {f : M → M'} (hf : ContinuousOn f s)
Goal: UniqueDiffOn 𝕜 ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
-- Trivialization.mdifferentiable
theorem Trivialization.mdifferentiable (e : Trivialization F (π F Z)) [MemTrivializationAtlas e] :
    e.toPartialHomeomorph.MDifferentiable (I.prod 𝓘(𝕜, F)) (I.prod 𝓘(𝕜, F)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {Z : M → Type u_9} [TopologicalSpace (Bundle.TotalSpace F Z)] [(b : M) → TopologicalSpace (Z b)] [(b : M) → AddCommMonoid (Z b)] [(b : M) → Module 𝕜 (Z b)] [FiberBundle F Z] [VectorBundle 𝕜 F Z] [SmoothVectorBundle F Z I] (e : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e]
Goal: PartialHomeomorph.MDifferentiable (I.prod 𝓘(𝕜, F)) (I.prod 𝓘(𝕜, F)) e.toPartialHomeomorph
-- UniqueMDiffWithinAt.smooth_bundle_preimage
theorem UniqueMDiffWithinAt.smooth_bundle_preimage {p : TotalSpace F Z}
    (hs : UniqueMDiffWithinAt I s p.proj) :
    UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (π F Z ⁻¹' s) p 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {s : Set M} {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {Z : M → Type u_9} [TopologicalSpace (Bundle.TotalSpace F Z)] [(b : M) → TopologicalSpace (Z b)] [(b : M) → AddCommMonoid (Z b)] [(b : M) → Module 𝕜 (Z b)] [FiberBundle F Z] [VectorBundle 𝕜 F Z] [SmoothVectorBundle F Z I] {p : Bundle.TotalSpace F Z} (hs : UniqueMDiffWithinAt I s p.proj)
Goal: UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (Bundle.TotalSpace.proj ⁻¹' s) p
-- UniqueMDiffWithinAt.smooth_bundle_preimage'
theorem UniqueMDiffWithinAt.smooth_bundle_preimage' {b : M} (hs : UniqueMDiffWithinAt I s b)
    (x : Z b) : UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (π F Z ⁻¹' s) ⟨b, x⟩ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {s : Set M} {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (Z : M → Type u_9) [TopologicalSpace (Bundle.TotalSpace F Z)] [(b : M) → TopologicalSpace (Z b)] [(b : M) → AddCommMonoid (Z b)] [(b : M) → Module 𝕜 (Z b)] [FiberBundle F Z] [VectorBundle 𝕜 F Z] [SmoothVectorBundle F Z I] {b : M} (hs : UniqueMDiffWithinAt I s b) (x : Z b)
Goal: UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (Bundle.TotalSpace.proj ⁻¹' s) { proj := b, snd := x }
-- UniqueMDiffOn.smooth_bundle_preimage
theorem UniqueMDiffOn.smooth_bundle_preimage (hs : UniqueMDiffOn I s) :
    UniqueMDiffOn (I.prod 𝓘(𝕜, F)) (π F Z ⁻¹' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {s : Set M} {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (Z : M → Type u_9) [TopologicalSpace (Bundle.TotalSpace F Z)] [(b : M) → TopologicalSpace (Z b)] [(b : M) → AddCommMonoid (Z b)] [(b : M) → Module 𝕜 (Z b)] [FiberBundle F Z] [VectorBundle 𝕜 F Z] [SmoothVectorBundle F Z I] (hs : UniqueMDiffOn I s)
Goal: UniqueMDiffOn (I.prod 𝓘(𝕜, F)) (Bundle.TotalSpace.proj ⁻¹' s)