Definition inj_on_list {A B} (l : list A) (f : A -> B) :=
  forall x y, In x l -> In y l -> f x = f y -> x = y.Definition subset_rel {X} (sub lst : list X) : Prop :=
  forall x,
    In x sub ->
    In x lst.

Lemma NoDup_disjointness: forall {A B} (l1 : list A) (l2 : list A) (f : A -> B) , NoDup (List.map f l1) -> NoDup (List.map f l2) -> disjoint_list (List.map f l1) (List.map f l2) -> NoDup (List.map f (l1 ++ l2)).