Definition pr1_invertible_2cell_total {a b : C} {f g : C⟦a,b⟧} {x : f ==> g} {aa : D a} {bb : D b} {ff : aa -->[ f ] bb} {gg : aa -->[ g ] bb} (xx : ff ==>[ x ] gg) : @is_invertible_2cell (total_bicat D) (a ,, aa) (b ,, bb) (f ,, ff) (g ,, gg) (x ,, xx) → is_invertible_2cell x. Proof. intros z. use tpair. - exact (pr11 z). - split. + exact (base_paths _ _ (pr12 z)). + exact (base_paths _ _ (pr22 z)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells.