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Theorem resindi 5162
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 5009 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
21ineq2i 3539 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
3 inindi 3558 . . 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 2456 . 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 4890 . 2  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
6 df-res 4890 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
7 df-res 4890 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
86, 7ineq12i 3540 . 2  |-  ( ( A  |`  B )  i^i  ( A  |`  C ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( A  i^i  ( C  X.  _V )
) )
94, 5, 83eqtr4i 2466 1  |-  ( A  |`  ( B  i^i  C
) )  =  ( ( A  |`  B )  i^i  ( A  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2956    i^i cin 3319    X. cxp 4876    |` cres 4880
This theorem is referenced by:  gsum2d  15546  resisresindm  28074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885  df-res 4890
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