Definition is_upper_bound_subfamily {I} (f : I -> P) {J} (g : J -> P) (ff : forall j, (image f) (g j)) {x} (x_ub : isupperbound f x) : isupperbound g x. Proof. intros j. refine (factor_through_squash_hProp _ _ (ff j)). intros [i e]; destruct e. apply x_ub. 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.