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Theorem resiun2 5222
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 4942 . 2  |-  ( C  |`  U_ x  e.  A  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
2 df-res 4942 . . . . 5  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
32a1i 11 . . . 4  |-  ( x  e.  A  ->  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) ) )
43iuneq2i 4279 . . 3  |-  U_ x  e.  A  ( C  |`  B )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
5 xpiundir 4986 . . . . 5  |-  ( U_ x  e.  A  B  X.  _V )  =  U_ x  e.  A  ( B  X.  _V )
65ineq2i 3628 . . . 4  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
7 iunin2 4324 . . . 4  |-  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
86, 7eqtr4i 2428 . . 3  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
94, 8eqtr4i 2428 . 2  |-  U_ x  e.  A  ( C  |`  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
101, 9eqtr4i 2428 1  |-  ( C  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    e. wcel 1836   _Vcvv 3051    i^i cin 3405   U_ciun 4260    X. cxp 4928    |` cres 4932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-iun 4262  df-opab 4443  df-xp 4936  df-res 4942
This theorem is referenced by:  fvn0ssdmfun  5941  dprd2da  17227
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