+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground/lib/subset.ma".
+
+(* EXTENSIONS FOR SUBSETS ***************************************************)
+
+definition subset_ext_f1 (A1) (A0) (f:A1→A0): 𝒫❨A1❩ → 𝒫❨A0❩ ≝
+ λu1,a0. ∃∃a1. a1 ϵ u1 & f a1 = a0.
+
+definition subset_ext_p1 (A1) (Q:predicate A1): predicate (𝒫❨A1❩) ≝
+ λu1. ∀a1. a1 ϵ u1 → Q a1.
+
+(* Basic constructions ******************************************************)
+
+lemma subset_in_ext_f1_dx (A1) (A0) (f) (u1) (a1):
+ a1 ϵ u1 → f a1 ϵ subset_ext_f1 A1 A0 f u1.
+/2 width=3 by ex2_intro/ qed.
+
+(* Basic inversions *********************************************************)
+
+lemma subset_in_inv_ext_p1_dx (A1) (Q) (u1) (a1):
+ a1 ϵ u1 → subset_ext_p1 A1 Q u1 → Q a1.
+/2 width=1 by/ qed-.