Theorem 2iunin 3932

Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

Assertion

Ref	Expression
2iunin	|- U_ x e. A U_ y e. B ( C i^i D ) = ( U_ x e. A C i^i U_ y e. B D )

Distinct variable groups:   x, B   y, C   x, D   x, y

Allowed substitution hints:   A( x, y)   B( y)   C( x)   D( y)

Proof of Theorem 2iunin

Step	Hyp	Ref	Expression
1		iunin2 3929	. . . 4 |- U_ y e. B ( C i^i D ) = ( C i^i U_ y e. B D )
2	1	a1i 9	. . 3 |- ( x e. A -> U_ y e. B ( C i^i D ) = ( C i^i U_ y e. B D ) )
3	2	iuneq2i 3884	. 2 |- U_ x e. A U_ y e. B ( C i^i D ) = U_ x e. A ( C i^i U_ y e. B D )
4		iunin1 3930	. 2 |- U_ x e. A ( C i^i U_ y e. B D ) = ( U_ x e. A C i^i U_ y e. B D )
5	3, 4	eqtri 2186	1 |- U_ x e. A U_ y e. B ( C i^i D ) = ( U_ x e. A C i^i U_ y e. B D )

Syntax hints:   = wceq 1343   e. wcel 2136   i^i cin 3115   U_ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iun 3868

This theorem is referenced by: (None)