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Theorem inres 4844
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 3291 . 2  |-  ( ( A  i^i  B )  i^i  ( C  X.  _V ) )  =  ( A  i^i  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4559 . 2  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  i^i  B
)  i^i  ( C  X.  _V ) )
3 df-res 4559 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
43ineq2i 3279 . 2  |-  ( A  i^i  ( B  |`  C ) )  =  ( A  i^i  ( B  i^i  ( C  X.  _V ) ) )
51, 2, 43eqtr4ri 2172 1  |-  ( A  i^i  ( B  |`  C ) )  =  ( ( A  i^i  B )  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1332   _Vcvv 2689    i^i cin 3075    X. cxp 4545    |` cres 4549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-res 4559
This theorem is referenced by:  resindm  4869
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