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

Lemma lem_fun_and_img :
forall f a b : E, fun_ f -> In (couple a b) f -> In b (Img f).