Section Image.
Variables U V : Type.
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.
Definition injective (f:U -> V) := forall x y:U, f x = f y -> x = y.

Lemma injective_preserves_cardinal :
forall (A:Ensemble U) (f:U -> V) (n:nat),
injective f ->
cardinal _ A n -> forall n':nat, cardinal _ (Im A f) n' -> n' = n.