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Theorem resiun2 4700
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 4423 . 2  |-  ( C  |`  U_ x  e.  A  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
2 df-res 4423 . . . . 5  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
32a1i 9 . . . 4  |-  ( x  e.  A  ->  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) ) )
43iuneq2i 3731 . . 3  |-  U_ x  e.  A  ( C  |`  B )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
5 xpiundir 4465 . . . . 5  |-  ( U_ x  e.  A  B  X.  _V )  =  U_ x  e.  A  ( B  X.  _V )
65ineq2i 3187 . . . 4  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
7 iunin2 3776 . . . 4  |-  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
86, 7eqtr4i 2108 . . 3  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
94, 8eqtr4i 2108 . 2  |-  U_ x  e.  A  ( C  |`  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
101, 9eqtr4i 2108 1  |-  ( C  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1287    e. wcel 1436   _Vcvv 2615    i^i cin 2987   U_ciun 3713    X. cxp 4409    |` cres 4413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-iun 3715  df-opab 3875  df-xp 4417  df-res 4423
This theorem is referenced by: (None)
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