proposition: Proposition isaprop_dependent_sum {C : cat_with_terminal_cleaving} {x y : C} (f : x --> y) : isaprop (dependent_sum (cleaving_of_types C) f).
proof: Proof. pose (D₁ := univalent_fiber_category (disp_cat_of_types C) y : bicat_of_univ_cats). pose (D₂ := univalent_fiber_category (disp_cat_of_types C) x : bicat_of_univ_cats). pose (F := (fiber_functor_from_cleaving _ (cleaving_of_types C) f : D₁ --> D₂)). use (isofhlevelweqf _ (right_adjoint_weq_is_right_adjoint F)). use (isofhlevelweqf _ op1_left_adjoint_weq_right_adjoint). apply isaprop_left_adjoint. use op1_bicat_is_univalent_2_1. exact univalent_cat_is_univalent_2_1. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.DependentSums UniMath.CategoryTheory.Limits.Pullbacks UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.AdjointUnique UniMath.Bicategories.Core.UnivalenceOp UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Examples.OpMorBicat UniMath.Bicategories.Morphisms.Adjunctions UniMath.Bicategories.Morphisms.Examples.MorphismsInBicatOfUnivCats UniMath.Bicategories.Morphisms.Examples.MorphismsInOp1Bicat UniMath.Bicategories.DisplayedBicats.DispBicat UniMath.Bicategories.DisplayedBicats.DispUnivalence UniMath.Bicategories.DisplayedBicats.Examples.LiftDispBicat UniMath.Bicategories.DisplayedBicats.Examples.FullSub UniMath.Bicategories.ComprehensionCat.BicatOfCompCat UniMath.Bicategories.ComprehensionCat.CompCatNotations