File: Mathlib/Topology/LocalAtTarget.lean

full_name: Set.restrictPreimage_inducing
code: theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
    Inducing (s.restrictPreimage f) 
goal: Inducing (s.restrictPreimage f)

full_name: Set.restrictPreimage_embedding
code: theorem Set.restrictPreimage_embedding (s : Set β) (h : Embedding f) :
    Embedding (s.restrictPreimage f) 
goal: Embedding (s.restrictPreimage f)

full_name: Set.restrictPreimage_openEmbedding
code: theorem Set.restrictPreimage_openEmbedding (s : Set β) (h : OpenEmbedding f) :
    OpenEmbedding (s.restrictPreimage f) 
goal: OpenEmbedding (s.restrictPreimage f)

full_name: Set.restrictPreimage_closedEmbedding
code: theorem Set.restrictPreimage_closedEmbedding (s : Set β) (h : ClosedEmbedding f) :
    ClosedEmbedding (s.restrictPreimage f) 
goal: ClosedEmbedding (s.restrictPreimage f)

full_name: IsClosedMap.restrictPreimage
code: theorem IsClosedMap.restrictPreimage (H : IsClosedMap f) (s : Set β) :
    IsClosedMap (s.restrictPreimage f) 
goal: IsClosedMap (s.restrictPreimage f)

full_name: Set.restrictPreimage_isClosedMap
code: @[deprecated (since := "2024-04-02")]
theorem Set.restrictPreimage_isClosedMap (s : Set β) (H : IsClosedMap f) :
    IsClosedMap (s.restrictPreimage f) 
goal: IsClosedMap (s.restrictPreimage f)

full_name: IsOpenMap.restrictPreimage
code: theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
    IsOpenMap (s.restrictPreimage f) 
goal: IsOpenMap (s.restrictPreimage f)

full_name: Set.restrictPreimage_isOpenMap
code: @[deprecated (since := "2024-04-02")]
theorem Set.restrictPreimage_isOpenMap (s : Set β) (H : IsOpenMap f) :
    IsOpenMap (s.restrictPreimage f) 
goal: IsOpenMap (s.restrictPreimage f)

full_name: isOpen_iff_inter_of_iSup_eq_top
code: theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) 
goal: IsOpen s ↔ ∀ (i : ι), IsOpen (s ∩ ↑(U i))

full_name: isOpen_iff_coe_preimage_of_iSup_eq_top
code: theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
    IsOpen s ↔ ∀ i, IsOpen ((↑) ⁻¹' s : Set (U i)) 
goal: IsOpen s ↔ ∀ (i : ι), IsOpen (Subtype.val ⁻¹' s)

full_name: isClosed_iff_coe_preimage_of_iSup_eq_top
code: theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
    IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) 
goal: IsClosed s ↔ ∀ (i : ι), IsClosed (Subtype.val ⁻¹' s)

full_name: isLocallyClosed_iff_coe_preimage_of_iSup_eq_top
code: theorem isLocallyClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
    IsLocallyClosed s ↔ ∀ i, IsLocallyClosed ((↑) ⁻¹' s : Set (U i)) 
goal: IsLocallyClosed s ↔ ∀ (i : ι), IsLocallyClosed (Subtype.val ⁻¹' s)

full_name: isOpenMap_iff_isOpenMap_of_iSup_eq_top
code: theorem isOpenMap_iff_isOpenMap_of_iSup_eq_top :
    IsOpenMap f ↔ ∀ i, IsOpenMap ((U i).1.restrictPreimage f) 
goal: IsOpenMap f ↔ ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)

full_name: isClosedMap_iff_isClosedMap_of_iSup_eq_top
code: theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
    IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) 
goal: IsClosedMap f ↔ ∀ (i : ι), IsClosedMap ((U i).carrier.restrictPreimage f)

full_name: inducing_iff_inducing_of_iSup_eq_top
code: theorem inducing_iff_inducing_of_iSup_eq_top (h : Continuous f) :
    Inducing f ↔ ∀ i, Inducing ((U i).1.restrictPreimage f) 
goal: Inducing f ↔ ∀ (i : ι), Inducing ((U i).carrier.restrictPreimage f)

full_name: embedding_iff_embedding_of_iSup_eq_top
code: theorem embedding_iff_embedding_of_iSup_eq_top (h : Continuous f) :
    Embedding f ↔ ∀ i, Embedding ((U i).1.restrictPreimage f) 
goal: Embedding f ↔ ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f)

full_name: openEmbedding_iff_openEmbedding_of_iSup_eq_top
code: theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
    OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) 
goal: OpenEmbedding f ↔ ∀ (i : ι), OpenEmbedding ((U i).carrier.restrictPreimage f)

full_name: closedEmbedding_iff_closedEmbedding_of_iSup_eq_top
code: theorem closedEmbedding_iff_closedEmbedding_of_iSup_eq_top (h : Continuous f) :
    ClosedEmbedding f ↔ ∀ i, ClosedEmbedding ((U i).1.restrictPreimage f) 
goal: ClosedEmbedding f ↔ ∀ (i : ι), ClosedEmbedding ((U i).carrier.restrictPreimage f)