unfold bijective in |- *.
intros A B f f' H' H'0; try assumption.
split; [ try assumption | idtac ].
elim H'; intros H'1 H'2; try exact H'1; clear H'.
apply injective_comp with (f := f); auto with algebra.
elim H'; intros H'1 H'2; try exact H'2; clear H'.
apply surjective_comp with (f := f); auto with algebra.
Qed.
Comments "Trivialities:".

Lemma bijective_injective :
forall (A B : Setoid) (f : MAP A B), bijective f -> injective f.