(* Proof of lem_fun_img_dir_is_in_pimg                                     *)
intros.
elim (axs_parties (Img f) (fun_img_directe f Af x)); intros; apply H1;
clear H0 H1; intros.
unfold fun_img_directe in H0.
elim
(axs_comprehension
(fun x0 : E =>
In x (parties (dom f)) -> exists y : E, In y x /\ x0 = App f Af y)
(Img f) v3); intros; generalize (H1 H0); clear H0 H1 H2;
intros.
elim H0; clear H0; intros; auto with zfc.
Qed.
(* End of proof of lem_fun_img_dir_is_in_pimg                              *)

Lemma lem_fun_img_prop2 :
forall (f : E) (Af : fun_ f) (x : E),
In x (parties (dom f)) ->
inc x (fun_img_inverse f Af (fun_img_directe f Af x)).