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Theorem resindi 4627
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 4472 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
21ineq2i 3135 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
3 inindi 3154 . . 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 2060 . 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 4357 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
6 df-res 4357 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
7 df-res 4357 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
86, 7ineq12i 3136 . 2  |-  ( ( A  |`  B )  i^i  ( A  |`  C ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( A  i^i  ( C  X.  _V )
) )
94, 5, 83eqtr4i 2070 1  |-  ( A  |`  ( B  i^i  C
) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1243   _Vcvv 2557    i^i cin 2916    X. cxp 4343    |` cres 4347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357
This theorem is referenced by: (None)
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