File: Mathlib/Topology/Bornology/Hom.lean

full_name: LocallyBoundedMap.toFun
code: /-- The type of bounded maps from `α` to `β`, the maps which send a bounded set to a bounded set. -/
structure LocallyBoundedMap (α β : Type*) [Bornology α] [Bornology β] where
  /-- The function underlying a locally bounded map -/
  toFun : α → β
  /-- The pullback of the `Bornology.cobounded` filter under the function is contained in the
  cobounded filter. Equivalently, the function maps bounded sets to bounded sets. -/
  comap_cobounded_le' : (cobounded β).comap toFun ≤ cobounded α
goal: α → β

full_name: LocallyBoundedMap.comap_cobounded_le'
code: /-- The type of bounded maps from `α` to `β`, the maps which send a bounded set to a bounded set. -/
structure LocallyBoundedMap (α β : Type*) [Bornology α] [Bornology β] where
  /-- The function underlying a locally bounded map -/
  toFun : α → β
  /-- The pullback of the `Bornology.cobounded` filter under the function is contained in the
  cobounded filter. Equivalently, the function maps bounded sets to bounded sets. -/
  comap_cobounded_le' : (cobounded β).comap toFun ≤ cobounded α
goal: Filter.comap self.toFun (Bornology.cobounded β) ≤ Bornology.cobounded α

full_name: LocallyBoundedMap
code: /-- The type of bounded maps from `α` to `β`, the maps which send a bounded set to a bounded set. -/
structure LocallyBoundedMap (α β : Type*) [Bornology α] [Bornology β] where
  /-- The function underlying a locally bounded map -/
  toFun : α → β
  /-- The pullback of the `Bornology.cobounded` filter under the function is contained in the
  cobounded filter. Equivalently, the function maps bounded sets to bounded sets. -/
  comap_cobounded_le' : (cobounded β).comap toFun ≤ cobounded α
goal: Type (max u_6 u_7)

full_name: LocallyBoundedMapClass
code: /-- `LocallyBoundedMapClass F α β` states that `F` is a type of bounded maps.

You should extend this class when you extend `LocallyBoundedMap`. -/
class LocallyBoundedMapClass (F : Type*) (α β : outParam Type*) [Bornology α]
    [Bornology β] [FunLike F α β] : Prop where
  /-- The pullback of the `Bornology.cobounded` filter under the function is contained in the
  cobounded filter. Equivalently, the function maps bounded sets to bounded sets. -/
  comap_cobounded_le (f : F) : (cobounded β).comap f ≤ cobounded α
goal: Prop

full_name: LocallyBoundedMapClass.comap_cobounded_le
code: /-- `LocallyBoundedMapClass F α β` states that `F` is a type of bounded maps.

You should extend this class when you extend `LocallyBoundedMap`. -/
class LocallyBoundedMapClass (F : Type*) (α β : outParam Type*) [Bornology α]
    [Bornology β] [FunLike F α β] : Prop where
  /-- The pullback of the `Bornology.cobounded` filter under the function is contained in the
  cobounded filter. Equivalently, the function maps bounded sets to bounded sets. -/
  comap_cobounded_le (f : F) : (cobounded β).comap f ≤ cobounded α
goal: Filter.comap (⇑f) (Bornology.cobounded β) ≤ Bornology.cobounded α

full_name: Bornology.IsBounded.image
code: theorem Bornology.IsBounded.image [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] (f : F)
    {s : Set α} (hs : IsBounded s) : IsBounded (f '' s) 
goal: Bornology.IsBounded (⇑f '' s)

full_name: LocallyBoundedMapClass.toLocallyBoundedMap
code: /-- Turn an element of a type `F` satisfying `LocallyBoundedMapClass F α β` into an actual
`LocallyBoundedMap`. This is declared as the default coercion from `F` to
`LocallyBoundedMap α β`. -/
@[coe]
def LocallyBoundedMapClass.toLocallyBoundedMap [Bornology α] [Bornology β]
    [LocallyBoundedMapClass F α β] (f : F) : LocallyBoundedMap α β 
goal: LocallyBoundedMap α β

full_name: instCoeTCLocallyBoundedMapOfLocallyBoundedMapClass
code: instance [Bornology α] [Bornology β] [LocallyBoundedMapClass F α β] :
    CoeTC F (LocallyBoundedMap α β) 
goal: CoeTC F (LocallyBoundedMap α β)

full_name: LocallyBoundedMap.instFunLike
code: instance : FunLike (LocallyBoundedMap α β) α β 
goal: FunLike (LocallyBoundedMap α β) α β

full_name: LocallyBoundedMap.instLocallyBoundedMapClass
code: instance : LocallyBoundedMapClass (LocallyBoundedMap α β) α β 
goal: LocallyBoundedMapClass (LocallyBoundedMap α β) α β

full_name: LocallyBoundedMap.ext
code: @[ext]
theorem ext {f g : LocallyBoundedMap α β} (h : ∀ a, f a = g a) : f = g 
goal: f = g

full_name: LocallyBoundedMap.copy
code: /-- Copy of a `LocallyBoundedMap` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = f) : LocallyBoundedMap α β 
goal: LocallyBoundedMap α β

full_name: LocallyBoundedMap.coe_copy
code: @[simp]
theorem coe_copy (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
goal: ⇑(f.copy f' h) = f'

full_name: LocallyBoundedMap.copy_eq
code: theorem copy_eq (f : LocallyBoundedMap α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
goal: f.copy f' h = f

full_name: LocallyBoundedMap.ofMapBounded
code: /-- Construct a `LocallyBoundedMap` from the fact that the function maps bounded sets to bounded
sets. -/
def ofMapBounded (f : α → β) (h : ∀ ⦃s : Set α⦄, IsBounded s → IsBounded (f '' s)) :
    LocallyBoundedMap α β 
goal: LocallyBoundedMap α β

full_name: LocallyBoundedMap.coe_ofMapBounded
code: @[simp]
theorem coe_ofMapBounded (f : α → β) {h} : ⇑(ofMapBounded f h) = f 
goal: ⇑(LocallyBoundedMap.ofMapBounded f h) = f

full_name: LocallyBoundedMap.ofMapBounded_apply
code: @[simp]
theorem ofMapBounded_apply (f : α → β) {h} (a : α) : ofMapBounded f h a = f a 
goal: (LocallyBoundedMap.ofMapBounded f h) a = f a

full_name: LocallyBoundedMap.id
code: /-- `id` as a `LocallyBoundedMap`. -/
protected def id : LocallyBoundedMap α α 
goal: LocallyBoundedMap α α

full_name: LocallyBoundedMap.instInhabited
code: instance : Inhabited (LocallyBoundedMap α α) 
goal: Inhabited (LocallyBoundedMap α α)

full_name: LocallyBoundedMap.coe_id
code: @[simp]
theorem coe_id : ⇑(LocallyBoundedMap.id α) = id 
goal: ⇑(LocallyBoundedMap.id α) = id

full_name: LocallyBoundedMap.id_apply
code: @[simp]
theorem id_apply (a : α) : LocallyBoundedMap.id α a = a 
goal: (LocallyBoundedMap.id α) a = a

full_name: LocallyBoundedMap.comp
code: /-- Composition of `LocallyBoundedMap`s as a `LocallyBoundedMap`. -/
def comp (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) : LocallyBoundedMap α γ 
goal: LocallyBoundedMap α γ

full_name: LocallyBoundedMap.coe_comp
code: @[simp]
theorem coe_comp (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) : ⇑(f.comp g) = f ∘ g 
goal: ⇑(f.comp g) = ⇑f ∘ ⇑g

full_name: LocallyBoundedMap.comp_apply
code: @[simp]
theorem comp_apply (f : LocallyBoundedMap β γ) (g : LocallyBoundedMap α β) (a : α) :
    f.comp g a = f (g a) 
goal: (f.comp g) a = f (g a)

full_name: LocallyBoundedMap.comp_assoc
code: @[simp]
theorem comp_assoc (f : LocallyBoundedMap γ δ) (g : LocallyBoundedMap β γ)
    (h : LocallyBoundedMap α β) : (f.comp g).comp h = f.comp (g.comp h) 
goal: (f.comp g).comp h = f.comp (g.comp h)

full_name: LocallyBoundedMap.comp_id
code: @[simp]
theorem comp_id (f : LocallyBoundedMap α β) : f.comp (LocallyBoundedMap.id α) = f 
goal: f.comp (LocallyBoundedMap.id α) = f

full_name: LocallyBoundedMap.id_comp
code: @[simp]
theorem id_comp (f : LocallyBoundedMap α β) : (LocallyBoundedMap.id β).comp f = f 
goal: (LocallyBoundedMap.id β).comp f = f

full_name: LocallyBoundedMap.cancel_right
code: @[simp]
theorem cancel_right {g₁ g₂ : LocallyBoundedMap β γ} {f : LocallyBoundedMap α β}
    (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂

full_name: LocallyBoundedMap.cancel_left
code: @[simp]
theorem cancel_left {g : LocallyBoundedMap β γ} {f₁ f₂ : LocallyBoundedMap α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂