equiv_path_TypeO@{i j} {fs : Funext} O (T T' : Type_@{i j} O) : (paths@{j} T.1 T'.1) <~> (T = T') := equiv_path_sigma_hprop@{j i j} T T'. Global Instance inO_TypeO {O : Subuniverse} (A : Type_ O) : In O A := A.2. Class MapIn (O : Subuniverse) {A B : Type} (f : A -> B) := inO_hfiber_ino_map : forall (b:B), In O (hfiber f b). Global Existing Instance inO_hfiber_ino_map. Section Subuniverse. Context (O : Subuniverse). Global Instance ishprop_mapinO `{Funext} {A B : Type} (f : A -> B) : IsHProp (MapIn O f). Proof. apply istrunc_forall. 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.