English
Membership in a range is equivalent to existence of a preimage: x ∈ Set.range f iff ∃ y, f y = x.
Русский
Членство x в образе множества равно существованию y с f y = x.
LaTeX
$$$$ x \in \operatorname{Set.range} f \quad\Leftrightarrow\quad \exists y, f y = x. $$$$
Lean4
/-- A subset `p : α → Prop` of the type closed under `ωSup` induces an
`OmegaCompletePartialOrder` on the subtype `{a : α // p a}`. -/
def subtype {α : Type*} [OmegaCompletePartialOrder α] (p : α → Prop) (hp : ∀ c : Chain α, (∀ i ∈ c, p i) → p (ωSup c)) :
OmegaCompletePartialOrder (Subtype p) :=
OmegaCompletePartialOrder.lift (OrderHom.Subtype.val p)
(fun c => ⟨ωSup _, hp (c.map (OrderHom.Subtype.val p)) fun _ ⟨n, q⟩ => q.symm ▸ (c n).2⟩) (fun _ _ h => h)
(fun _ => rfl)
