{A : Type} (B : A -> Type) : (forall a, In O (B a)) <-> MapIn O (@pr1 A B). Proof. split. - exact _. - rapply functor_forall; intros a x. exact (inO_equiv_inO (hfiber pr1 a) (hfiber_fibration a B)^-1%equiv). Defined.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback. Require Import Homotopy.IdentitySystems. Require Import Colimits.Pushout.