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Theorem 2iunin 4360
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
StepHypRef Expression
1 iunin2 4356 . . . 4  |-  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
21a1i 11 . . 3  |-  ( x  e.  A  ->  U_ y  e.  B  ( C  i^i  D )  =  ( C  i^i  U_ y  e.  B  D )
)
32iuneq2i 4311 . 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 4357 . 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 )
53, 4eqtri 2484 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 )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455    e. wcel 1898    i^i cin 3415   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-iun 4294
This theorem is referenced by:  fpar  6927
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