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Theorem resiun2 5130
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 4851 . 2  |-  ( C  |`  U_ x  e.  A  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
2 df-res 4851 . . . . 5  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
32a1i 11 . . . 4  |-  ( x  e.  A  ->  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) ) )
43iuneq2i 4288 . . 3  |-  U_ x  e.  A  ( C  |`  B )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
5 xpiundir 4895 . . . . 5  |-  ( U_ x  e.  A  B  X.  _V )  =  U_ x  e.  A  ( B  X.  _V )
65ineq2i 3622 . . . 4  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
7 iunin2 4333 . . . 4  |-  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )  =  ( C  i^i  U_ x  e.  A  ( B  X.  _V ) )
86, 7eqtr4i 2496 . . 3  |-  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  U_ x  e.  A  ( C  i^i  ( B  X.  _V ) )
94, 8eqtr4i 2496 . 2  |-  U_ x  e.  A  ( C  |`  B )  =  ( C  i^i  ( U_ x  e.  A  B  X.  _V ) )
101, 9eqtr4i 2496 1  |-  ( C  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( C  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389   U_ciun 4269    X. cxp 4837    |` cres 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4271  df-opab 4455  df-xp 4845  df-res 4851
This theorem is referenced by:  fvn0ssdmfun  6028  dprd2da  17753
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