Inductive Im (X:Ensemble U) (f:U -> V) : Ensemble V :=
    Im_intro : forall x:U, In _ X x -> forall y:V, y = f x -> In _ (Im X f) y.Inductive Im (X:Ensemble U) (f:U -> V) : Ensemble V :=
    Im_intro : forall x:U, In _ X x -> forall y:V, y = f x -> In _ (Im X f) y.

Lemma Im_inv : forall (X:Ensemble U) (f:U -> V) (y:V), In _ (Im X f) y -> exists x : U, In _ X x /\ f x = y.