Theorem 2iunin 3983

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 3980	. . . 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 3934	. 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 3981	. 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 2217	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 1364   e. wcel 2167   i^i cin 3156   U_ciun 3916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-iun 3918

This theorem is referenced by: (None)