Definition disp_univalent_2_0_fullsubbicat (HC : is_univalent_2 C) (HP : ∏ (x : C), isaprop (P x)) : disp_univalent_2_0 disp_fullsubbicat. Proof. intros x y p xx yy. induction p. use isweqimplimpl. - intros ; cbn in *. apply HP. - apply isasetaprop. apply HP. - simple refine (isaprop_total2 (_ ,, _) (λ η , _ ,, _)). + exact isapropunit. + apply isaprop_disp_left_adjoint_equivalence. * apply (pr2 HC). * exact disp_fullsubbicat_univalent_2_1. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.Initial. Require Import UniMath.Bicategories.Core.Examples.Final. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.DisplayedCatToBicat.