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Theorem resiun2 4809
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 4521 . 2  |-  ( C  |`  U_ x  e.  A  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
2 df-res 4521 . . . . 5  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
32a1i 9 . . . 4  |-  ( x  e.  A  ->  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) ) )
43iuneq2i 3801 . . 3  |-  U_ x  e.  A  ( C  |`  B )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
5 xpiundir 4568 . . . . 5  |-  ( U_ x  e.  A  B  X.  _V )  =  U_ x  e.  A  ( B  X.  _V )
65ineq2i 3244 . . . 4  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
7 iunin2 3846 . . . 4  |-  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
86, 7eqtr4i 2141 . . 3  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
94, 8eqtr4i 2141 . 2  |-  U_ x  e.  A  ( C  |`  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
101, 9eqtr4i 2141 1  |-  ( C  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   _Vcvv 2660    i^i cin 3040   U_ciun 3783    X. cxp 4507    |` cres 4511
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iun 3785  df-opab 3960  df-xp 4515  df-res 4521
This theorem is referenced by: (None)
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