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 finite_image : forall (X:Ensemble U) (f:U -> V), Finite _ X -> Finite _ (Im X f).