Definition transportf_subtypePath {A : UU} {P : A → UU} (Pprop : ∏ (a : A), isaprop (P a)) {C : total2 P → UU} (x : A) (P₁ P₂ : P x) (y : C (x,,P₂)) : transportf (λ (z : total2 P), C z) (!(@subtypePath A P Pprop (x ,, P₁) (x ,, P₂) (idpath x))) y = transportf (λ (p : P x), C (x,, p)) (!(pr1 (Pprop x P₁ P₂))) y. Proof. cbn. induction (Pprop x P₁ P₂) as [p q]. induction p. 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.