Lemma disp_full_sub_univalent (C : category) (P : C → UU) : (∏ x : C, isaprop (P x)) → is_univalent_disp (disp_full_sub C P). Proof. intro HP. apply is_univalent_disp_from_fibers. intros x xx xx'. cbn in *. apply isweqcontrprop. apply HP. apply isofhleveltotal2. - apply isapropunit. - intro. apply (@isaprop_is_z_iso_disp _ (disp_full_sub C P)). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.