proposition: Lemma in_on2P : {in D1 &, {on D2 &, allQ2 f}} <-> {in [pred x in D1 | f x \in D2] &, allQ2 f}.
proof: Proof. split => allf x y; have := allf x y; rewrite !inE => Q2f. by move=> /andP[? ?] /andP[? ?]; apply: Q2f. by move=> ? ? ? ?; apply: Q2f; apply/andP. Qed.
imports: ssreflect ssrfun