Theorem uniin1 28213

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

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem uniin1

Step	Hyp	Ref	Expression
1		iunin1 4357	. 2 |- U_ x e. A ( x i^i B ) = ( U_ x e. A x i^i B )
2		uniiun 4345	. . 3 |- U. A = U_ x e. A x
3	2	ineq1i 3642	. 2 |- ( U. A i^i B ) = ( U_ x e. A x i^i B )
4	1, 3	eqtr4i 2487	1 |- U_ x e. A ( x i^i B ) = ( U. A i^i B )

Syntax hints:   = wceq 1455   i^i cin 3415   U.cuni 4212   U_ciun 4292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-rex 2755  df-v 3059  df-in 3423  df-ss 3430  df-uni 4213  df-iun 4294

This theorem is referenced by: (None)