Instance eq_bigcap T U : Proper (set_eq ==> pointwise_relation T (@set_eq U) ==> set_eq) bigcap.

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)).