Lemma perm_map_in : forall (A B : Set)(lsa : list A)(lsb : list B)(f : A -> B)(f_inv : B -> A), (forall b, In b lsb -> f (f_inv b) = b) -> (forall a, In a lsa <-> In (f a) lsb) -> NoDup (map f lsa) -> NoDup lsb -> Permutation (map f lsa) lsb.