Lemmas about images of intervals under order isomorphisms.

@[simp]

theorem OrderIso.preimage_Iic {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β) : ⇑e ⁻¹' Set.Iic b = Set.Iic ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Ici {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β) : ⇑e ⁻¹' Set.Ici b = Set.Ici ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Iio {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β) : ⇑e ⁻¹' Set.Iio b = Set.Iio ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Ioi {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (b : β) : ⇑e ⁻¹' Set.Ioi b = Set.Ioi ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Icc {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : β) (b : β) : ⇑e ⁻¹' Set.Icc a b = Set.Icc ((OrderIso.symm e) a) ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Ico {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : β) (b : β) : ⇑e ⁻¹' Set.Ico a b = Set.Ico ((OrderIso.symm e) a) ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Ioc {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : β) (b : β) : ⇑e ⁻¹' Set.Ioc a b = Set.Ioc ((OrderIso.symm e) a) ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.preimage_Ioo {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : β) (b : β) : ⇑e ⁻¹' Set.Ioo a b = Set.Ioo ((OrderIso.symm e) a) ((OrderIso.symm e) b)

@[simp]

theorem OrderIso.image_Iic {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α) : ⇑e '' Set.Iic a = Set.Iic (e a)

@[simp]

theorem OrderIso.image_Ici {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α) : ⇑e '' Set.Ici a = Set.Ici (e a)

@[simp]

theorem OrderIso.image_Iio {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α) : ⇑e '' Set.Iio a = Set.Iio (e a)

@[simp]

theorem OrderIso.image_Ioi {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) (a : α) : ⇑e '' Set.Ioi a = Set.Ioi (e a)

@[simp]

theorem OrderIso.image_Ioo {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (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} [Preorder α] [Preorder β] (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} [Preorder α] [Preorder β] (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} [Preorder α] [Preorder β] (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} [Preorder α] [OrderTop α] : ↑(Set.Iic ⊤) ≃o α

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

Equations

• OrderIso.IicTop = let __src := Equiv.subtypeUnivEquiv ⋯; { toEquiv := __src, map_rel_iff' := ⋯ }

def OrderIso.IciBot {α : Type u_1} [Preorder α] [OrderBot α] : ↑(Set.Ici ⊥) ≃o α

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

Equations

• OrderIso.IciBot = let __src := Equiv.subtypeUnivEquiv ⋯; { toEquiv := __src, map_rel_iff' := ⋯ }