proposition: Lemma iff_forall_inO_mapinO_pr1 {A : Type} (B : A -> Type) : (forall a, In O (B a)) <-> MapIn O (@pr1 A B).
proof: Proof. split. - exact _. - rapply functor_forall; intros a x. exact (inO_equiv_inO (hfiber pr1 a) (hfiber_fibration a B)^-1%equiv). Defined.
imports: HoTT.Basics HoTT.Types Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback PathAny Colimits.Pushout