Theorem uniin1

Description: Union of intersection. Generalization of half of theorem "Distributive laws" in Enderton p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020)

	Ref	Expression
Assertion	uniin1	|- U_ x e. A ( x i^i B ) = ( U. A i^i B )

Proof

Step	Hyp	Ref	Expression
1		iunin1	|- U_ x e. A ( x i^i B ) = ( U_ x e. A x i^i B )
2		uniiun	|- U. A = U_ x e. A x
3	2	ineq1i	|- ( U. A i^i B ) = ( U_ x e. A x i^i B )
4	1 3	eqtr4i	|- U_ x e. A ( x i^i B ) = ( U. A i^i B )