(* Proof of lem_rel_and_rdom2                                              *)
unfold rdom2 in |- *; intros.
elim
(axs_comprehension (fun x0 : E => exists y : E, In (couple x0 y) r)
(reunion (reunion r)) x); intros; apply H1; clear H0 H1;
intros.
split; [ idtac | exists y; auto with zfc ].
elim (axs_reunion (reunion r) x); intros; apply H1; clear H0 H1; intros.
exists (singleton x); split; [ idtac | apply lem_x_in_sing_x ].
elim (axs_reunion r (singleton x)); intros; apply H1; clear H0 H1; intros.
exists (couple x y); split; [ auto with zfc | idtac ].
elim (axs_paire (singleton x) (paire x y) (singleton x)); intros; apply H1;
clear H0 H1; intros.
left; reflexivity.
Qed.
(* End of proof of lem_rel_and_rdom2                                       *)

(***************************************************************************)
(* Si (x,y) est dans r alors y est dans l'image de r.                      *)
(***************************************************************************)
Lemma lem_rel_and_rImg :
forall (r a b : E) (rRel : rel_sig r a b) (x y : E),
In (couple x y) r -> In y (rImg r a b rRel).