Definition sigma_disp_univalent_2_1_with_props : disp_univalent_2_1 (sigma_bicat _ _ D₂). Proof. apply fiberwise_local_univalent_is_univalent_2_1. intros x y f xx yy ff gg. use weqhomot. - cbn. refine (_ ∘ total2_paths_equiv _ _ _)%weq. refine (pair_disp_invertible_to_sigma_disp_invertible_weq _ _ ∘ _)%weq. induction ff as [ff1 ff2] ; induction gg as [gg1 gg2]. refine (weqtotal2 (make_weq _ (HD₁_2_1 _ _ _ _ (idpath _) _ _ ff1 gg1)) _). intro p ; cbn in p. induction p. exact (make_weq _ (HD₂_2_1 _ _ _ _ (idpath _) _ _ ff2 gg2)). - intros p. cbn in p. induction p. use subtypePath. { intro ; apply isaprop_is_disp_invertible_2cell. } apply idpath. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. 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.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Core.AdjointUnique. Require Import UniMath.Bicategories.DisplayedBicats.DispInvertibles. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence.