(* Proof of lem_fun_and_img                                                *)
intros; unfold Img in |- *.
elim
(axs_comprehension
(fun v1 : E => fun_ f -> exists v0 : E, In (couple v0 v1) f)
(reunion (reunion f)) b); intros; apply H2; clear H1 H2;
intros.
split; [ idtac | intros; exists a; auto with zfc ].
elim (axs_reunion (reunion f) b); intros; apply H2; clear H1 H2; intros;
exists (paire a b).
split;
[ idtac | elim (axs_paire a b b); intros; apply H2; right; reflexivity ].
elim (axs_reunion f (paire a b)); intros; apply H2; clear H1 H2; intros.
exists (couple a b); split;
[ auto with zfc
| unfold couple in |- *;
elim (axs_paire (singleton a) (paire a b) (paire a b));
intros; apply H2; right; reflexivity ].
Qed.
(* End of proof of lem_fun_and_img                                         *)

(***************************************************************************)
(* Si f est une application et x est un element du domaine de f alors il   *)
(* existe un y tel que (x,y) in f                                          *)
(***************************************************************************)
Lemma lem_App :
forall f x : E,
fun_ f -> In x (dom f) -> exists y : E, In y (Img f) /\ In (couple x y) f.