Lemma isaprop_po_sieve_dom (y : X) (f : sieve (C := po_category P) x) : isaprop (sieve_functor f y : hSet). Proof. apply invproofirrelevance. intros z w. apply (invmaponpathsincl _ (presheaf_monic_isincl (Subobject_isM f) _)). apply propproperty. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.GrothendieckToposes.Sieves. Require Import UniMath.CategoryTheory.Categories.PreorderCategory.Core. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Subobjects. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.OrderTheory.Preorders. Require Import UniMath.CategoryTheory.yoneda. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.slicecat.