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Theorem resindi 4921
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 4762 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
21ineq2i 3333 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
3 inindi 3352 . . 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 2198 . 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 4637 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
6 df-res 4637 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
7 df-res 4637 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
86, 7ineq12i 3334 . 2  |-  ( ( A  |`  B )  i^i  ( A  |`  C ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( A  i^i  ( C  X.  _V )
) )
94, 5, 83eqtr4i 2208 1  |-  ( A  |`  ( B  i^i  C
) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1353   _Vcvv 2737    i^i cin 3128    X. cxp 4623    |` cres 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4064  df-xp 4631  df-rel 4632  df-res 4637
This theorem is referenced by:  resindm  4948
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