(* Pas de reciproque (dixit pagano pb des ensembles non decidables)        *)
(* Proof of lem_subset                                                     *)
intros; elim (axs_comprehension F b y); intros.
generalize H1; rewrite <- H; clear H H1 H2; intros.
elim H1; intros; auto with zfc.
Qed.
(* End of proof of lem_subset                                              *)
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(* applications                                                            *)
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Definition fun_ (f : E) :=
(exists a : E, (exists b : E, inc f (cartesien a b))) /\
(forall x y z : E, In (couple x y) f -> In (couple x z) f -> y = z).
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(* Domaine d'une application                                               *)
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Definition dom (f : E) :=
subset (fun v0 : E => fun_ f -> exists v1 : E, In (couple v0 v1) f)
(reunion (reunion f)).
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(* Image d'une application                                                 *)
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Definition Img (f : E) :=
subset (fun v1 : E => fun_ f -> exists v0 : E, In (couple v0 v1) f)
(reunion (reunion f)).
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(* f est une application de a vers b                                       *)
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Definition fun_sig (f a b : E) := fun_ f /\ dom f = a /\ inc (Img f) b.

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(* Trois lemmes evidents mais utiles                                       *)
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Lemma lem_fun_sig_is_fun : forall f a b : E, fun_sig f a b -> fun_ f.
unfold fun_sig in |- *; intros.