Definition image_directed {P:Poset} : Directed_family P -> Directed_hsubtype P. Proof. intros p. exists (image p). apply image_is_directed, directed_property. 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.