Definition disp_depprod_bicat_disp_is_invertible_2cell_map {a b : B} {f g : B ⟦ a, b ⟧} {x : f ==> g} {xinv : is_invertible_2cell x} {aa : disp_depprod_bicat a} {bb : disp_depprod_bicat b} {ff : aa -->[ f] bb} {gg : aa -->[ g ] bb} (xx : ff ==>[ x ] gg) (Hx : ∏ (i : I), is_disp_invertible_2cell xinv (xx i)) : is_disp_invertible_2cell xinv xx. Proof. use tpair. - exact (λ i, pr1 (Hx i)). - split. + use funextsec. intro i. refine (_ @ disp_depprod_prebicat_laws_help _ _ _). exact (disp_vcomp_rinv ((xx i ,, Hx i) : disp_invertible_2cell (x ,, xinv) _ _)). + use funextsec. intro i. refine (_ @ disp_depprod_prebicat_laws_help _ _ _). exact (disp_vcomp_linv ((xx i ,, Hx i) : disp_invertible_2cell (x ,, xinv) _ _)). 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.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Univalence.