Coercion seq_In T : seq T -> set T := fun s x => List.In x s.Coercion list_In T : list T -> set T := fun s x => List.In x s.

Lemma bigcup_setI_cons_subset_r : forall (A B : Type) (l : A) (ls : seq A) (f : A -> set B) (s1 s2 : set B) (s3 : set A), s3 l -> s1 \subset f l -> s2 \subset \bigcup_(x in ls :&: s3) f x -> s1 :|: s2 \subset \bigcup_(x in (l :: ls) :&: s3) f x.