Definition image_is_chain {P:Poset} {I:UU} (p : I -> P) : is_chain p -> is_chain (pr1carrier (image p)). Proof. intros p_chain. unfold is_chain. apply (image_carrier_univ p (fun x => ∀ j, comparable x _)); intros i. apply (image_carrier_univ p); intros j. use p_chain. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.OrderedSets.WellOrderedSets. Require Import UniMath.OrderTheory.DCPOs.AlternativeDefinitions.Dcpo. Require Import UniMath.OrderTheory.Posets.Subposets.