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Theorem inres 4772
Description: Move intersection into class restriction. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
inres  |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )

Proof of Theorem inres
StepHypRef Expression
1 inass 3233 . 2  |-  ( ( A  i^i  B )  i^i  ( C  X.  _V ) )  =  ( A  i^i  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4489 . 2  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  i^i  B
)  i^i  ( C  X.  _V ) )
3 df-res 4489 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
43ineq2i 3221 . 2  |-  ( A  i^i  ( B  |`  C ) )  =  ( A  i^i  ( B  i^i  ( C  X.  _V ) ) )
51, 2, 43eqtr4ri 2131 1  |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1299   _Vcvv 2641    i^i cin 3020    X. cxp 4475    |` cres 4479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-res 4489
This theorem is referenced by:  resindm  4797
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