English
There is a canonical lifting of a bijection to an equivalence, i.e., there is a natural bijection between bijections α → β and equivalences α ≃ β, realized by the map f ↦ ofBijective f with inverse given by the underlying bijection of an equivalence.
Русский
Существует каноническое восхождение биекции к эквиваленции: естественная биекция между биекциями α → β и эквивалентностями α ≃ β, реализованная отображением f ↦ ofBijective f и обратная через базовую биекцию эквивалентности.
LaTeX
$$$\\{ f: α \\to β \\mid \\text{Bijective}(f) \\} \\cong \\mathrm{Equiv}(α,β),$ \\\\ $f \\mapsto \\text{ofBijective}(f)$, \\\\ and the inverse sends an equivalence to its underlying bijection.$$
Lean4
instance : CanLift (α → β) (α ≃ β) (↑) Bijective where prf f hf := ⟨ofBijective f hf, rfl⟩
