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_set_U {A B} (s1 s2 : set A) (s3 : set B) (f : A -> set B) : \bigcup_(x in s1) (f x) \subset s3 -> \bigcup_(x in s2) (f x) \subset s3 -> \bigcup_(x in (s1 :|: s2)) (f x) \subset s3.