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 lift_bigcup_comm : forall (U T : Type) (s : set U) (f : U -> set T), inhabited U -> lift (\bigcup_(i in [set : U]) (f i)) <--> \bigcup_(i in [set : U]) (lift (f i)).