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Theorem resindi 4971
Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
resindi  |-  ( A  |`  ( B  i^i  C
) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )

Proof of Theorem resindi
StepHypRef Expression
1 xpindir 4820 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
21ineq2i 3367 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
3 inindi 3386 . . 3  |-  ( A  i^i  ( ( B  X.  _V )  i^i  ( C  X.  _V ) ) )  =  ( ( A  i^i  ( B  X.  _V )
)  i^i  ( A  i^i  ( C  X.  _V ) ) )
42, 3eqtri 2303 . 2  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( ( A  i^i  ( B  X.  _V )
)  i^i  ( A  i^i  ( C  X.  _V ) ) )
5 df-res 4701 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
6 df-res 4701 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
7 df-res 4701 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
86, 7ineq12i 3368 . 2  |-  ( ( A  |`  B )  i^i  ( A  |`  C ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( A  i^i  ( C  X.  _V )
) )
94, 5, 83eqtr4i 2313 1  |-  ( A  |`  ( B  i^i  C
) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151    X. cxp 4687    |` cres 4691
This theorem is referenced by:  gsum2d  15223  int2pre  25237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-res 4701
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