Definition fmap_is_directed {P Q} (f : posetmorphism P Q) {I : UU} (p : I -> P) : isdirected p -> isdirected (f ∘ p). Proof. intros p_directed; split. - eapply isdirected_inhabited, p_directed. - intros i j. eapply exists_monotone. 2: { exact (isdirected_compatible p_directed i j). } intros x [le_i_x le_j_x]. split; apply posetmorphism_property; assumption. 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.