Arguments eq_bigcupl [T U A] B F _ _.
Global Instance eq_bigcup T U : Proper (set_eq ==> pointwise_relation T (@set_eq U) ==> set_eq) bigcup.

(* Specifications: *)
Lemma bigcup_flatten T U V A (F : T -> set U) (G : U -> set V) :
\bigcup_(x in \bigcup_(y in A) F y) G x <-->
\bigcup_(y in A) \bigcup_(x in F y) G x.
Lemma codom_apply {A B : Type} {f : A -> B} {x : A} : f x \in codom f.
Lemma codom_comp T U V (f : U -> T) (g : V -> U) :
codom (f \o g) <--> f @: (codom g).
Lemma curry_imset2r T U V (f : T -> U -> V) A1 A2 :
f @2: (A1, A2) <--> \bigcup_(x2 in A2) f^~ x2 @: A1.
Lemma imset_bigcup T U V (f : U -> V) A (F : T -> set U) :
(f @: \bigcup_(x in A) (F x)) <--> \bigcup_(x in A) f @: F x.
Lemma bigcup_codom T U V (f : T -> U) (F : U -> set V) :
\bigcup_(y in codom f) (F y) <--> \bigcup_x F (f x).
Lemma reindex_bigcup I J K (h : J -> I) (F : I -> set K) A B :
h @: B <--> A ->
\bigcup_(x in A) F x <--> \bigcup_(y in B) F (h y).