Definition is_lower_bound_subfamily {I} (f : I -> P) {J} (g : J -> P) (ff : forall j, (image f) (g j)) {x} (x_lb : islowerbound f x) : islowerbound g x. Proof. intros j. refine (factor_through_squash_hProp _ _ (ff j)). intros [i e]; destruct e. use x_lb. 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.