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Theorem resiun2 4991
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun2  |-  ( C  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  |`  B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem resiun2
StepHypRef Expression
1 df-res 4717 . 2  |-  ( C  |`  U_ x  e.  A  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
2 df-res 4717 . . . . 5  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
32a1i 10 . . . 4  |-  ( x  e.  A  ->  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) ) )
43iuneq2i 3939 . . 3  |-  U_ x  e.  A  ( C  |`  B )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
5 xpiundir 4760 . . . . 5  |-  ( U_ x  e.  A  B  X.  _V )  =  U_ x  e.  A  ( B  X.  _V )
65ineq2i 3380 . . . 4  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
7 iunin2 3982 . . . 4  |-  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
86, 7eqtr4i 2319 . . 3  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
94, 8eqtr4i 2319 . 2  |-  U_ x  e.  A  ( C  |`  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
101, 9eqtr4i 2319 1  |-  ( C  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   U_ciun 3921    X. cxp 4703    |` cres 4707
This theorem is referenced by:  dprd2da  15293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-iun 3923  df-opab 4094  df-xp 4711  df-res 4717
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