Theorem 2iunin 3887

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 3884	. . . 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 3839	. 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 3885	. 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 2161	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 1332   e. wcel 1481   i^i cin 3075   U_ciun 3821

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-iun 3823

This theorem is referenced by: (None)