# Lemmas about images of intervals under order isomorphisms. #

@[simp]
theorem OrderIso.preimage_Iic {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (b : β) :
e ⁻¹' = Set.Iic (() b)
@[simp]
theorem OrderIso.preimage_Ici {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (b : β) :
e ⁻¹' = Set.Ici (() b)
@[simp]
theorem OrderIso.preimage_Iio {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (b : β) :
e ⁻¹' = Set.Iio (() b)
@[simp]
theorem OrderIso.preimage_Ioi {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (b : β) :
e ⁻¹' = Set.Ioi (() b)
@[simp]
theorem OrderIso.preimage_Icc {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : β) (b : β) :
e ⁻¹' Set.Icc a b = Set.Icc (() a) (() b)
@[simp]
theorem OrderIso.preimage_Ico {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : β) (b : β) :
e ⁻¹' Set.Ico a b = Set.Ico (() a) (() b)
@[simp]
theorem OrderIso.preimage_Ioc {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : β) (b : β) :
e ⁻¹' Set.Ioc a b = Set.Ioc (() a) (() b)
@[simp]
theorem OrderIso.preimage_Ioo {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : β) (b : β) :
e ⁻¹' Set.Ioo a b = Set.Ioo (() a) (() b)
@[simp]
theorem OrderIso.image_Iic {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) :
e '' = Set.Iic (e a)
@[simp]
theorem OrderIso.image_Ici {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) :
e '' = Set.Ici (e a)
@[simp]
theorem OrderIso.image_Iio {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) :
e '' = Set.Iio (e a)
@[simp]
theorem OrderIso.image_Ioi {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) :
e '' = Set.Ioi (e a)
@[simp]
theorem OrderIso.image_Ioo {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) (b : α) :
e '' Set.Ioo a b = Set.Ioo (e a) (e b)
@[simp]
theorem OrderIso.image_Ioc {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) (b : α) :
e '' Set.Ioc a b = Set.Ioc (e a) (e b)
@[simp]
theorem OrderIso.image_Ico {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) (b : α) :
e '' Set.Ico a b = Set.Ico (e a) (e b)
@[simp]
theorem OrderIso.image_Icc {α : Type u_1} {β : Type u_2} [] [] (e : α ≃o β) (a : α) (b : α) :
e '' Set.Icc a b = Set.Icc (e a) (e b)
def OrderIso.IicTop {α : Type u_1} [] [] :
() ≃o α

Order isomorphism between Iic (⊤ : α) and α when α has a top element

Equations
• OrderIso.IicTop = let __src := ; { toEquiv := __src, map_rel_iff' := }
Instances For
def OrderIso.IciBot {α : Type u_1} [] [] :
() ≃o α

Order isomorphism between Ici (⊥ : α) and α when α has a bottom element

Equations
• OrderIso.IciBot = let __src := ; { toEquiv := __src, map_rel_iff' := }
Instances For