Definition is_least_upper_bound_image {I} (f : I -> P) {x : P} : is_least_upper_bound f x <-> is_least_upper_bound (pr1carrier (image f)) x. Proof. revert x. apply hconj_equiv. - use image_same_upper_bounds. - apply islowerbound_subtype_equiv. use image_same_upper_bounds. 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.