Lemma isaprop_is_strict_bicat (B : bicat) : isaprop (is_strict_bicat B). Proof. apply invproofirrelevance. intros x y. induction x as [LSx CSx]. induction y as [LSY CSy]. use pathsdirprod. - do 2 (use funextsec ; intro). apply isapropisaset. - use subtypePath. { exact (isPredicate_is_coh_strictness_structure_from_locally_strict LSx). } use subtypePath. { exact (isPredicate_strictness_structure_laws B). } repeat use pathsdirprod. + repeat (use funextsec ; intro). apply LSx. + repeat (use funextsec ; intro). apply LSx. + repeat (use funextsec ; intro). apply LSx. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.TwoCategories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Categories.HSET.All. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.TransportLaws.