Theorem 2iunin 3939

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 3936	. . . 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 3891	. 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 3937	. 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 2191	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 1348   e. wcel 2141   i^i cin 3120   U_ciun 3873

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875

This theorem is referenced by: (None)