Definition bicat_is_invertible_2cell_is_fullsub_is_invertible_2cell {X Y : fullsubbicat} {f g : X --> Y} (α : f ==> g) : @is_invertible_2cell C (pr1 X) (pr1 Y) (pr1 f) (pr1 g) (pr1 α) ≃ is_invertible_2cell α. Proof. use weqimplimpl. - exact (bicat_is_invertible_2cell_to_fullsub_is_invertible_2cell α). - exact (fullsub_is_invertible_2cell_to_bicat_is_invertible_2cell α). - apply isaprop_is_invertible_2cell. - apply isaprop_is_invertible_2cell. 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.