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Theorem resindir 5165
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 3561 . 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 4892 . 2  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  i^i  B
)  i^i  ( C  X.  _V ) )
3 df-res 4892 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
4 df-res 4892 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
53, 4ineq12i 3542 . 2  |-  ( ( A  |`  C )  i^i  ( B  |`  C ) )  =  ( ( A  i^i  ( C  X.  _V ) )  i^i  ( B  i^i  ( C  X.  _V )
) )
61, 2, 53eqtr4i 2468 1  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    i^i cin 3321    X. cxp 4878    |` cres 4882
This theorem is referenced by:  inimass  5290  fnreseql  5842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-res 4892
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