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Theorem resindir 4972
Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
resindir  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )

Proof of Theorem resindir
StepHypRef Expression
1 inindir 3387 . 2  |-  ( ( A  i^i  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( C  X.  _V ) )  i^i  ( B  i^i  ( C  X.  _V )
) )
2 df-res 4701 . 2  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  i^i  B
)  i^i  ( C  X.  _V ) )
3 df-res 4701 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
4 df-res 4701 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
53, 4ineq12i 3368 . 2  |-  ( ( A  |`  C )  i^i  ( B  |`  C ) )  =  ( ( A  i^i  ( C  X.  _V ) )  i^i  ( B  i^i  ( C  X.  _V )
) )
61, 2, 53eqtr4i 2313 1  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  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:  fnreseql  5635
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2790  df-in 3159  df-res 4701
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