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Theorem resres 4683
Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
resres  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )

Proof of Theorem resres
StepHypRef Expression
1 df-res 4413 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  B )  i^i  ( C  X.  _V ) )
2 df-res 4413 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
32ineq1i 3181 . 2  |-  ( ( A  |`  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )
4 xpindir 4530 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
54ineq2i 3182 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
6 df-res 4413 . . 3  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
7 inass 3194 . . 3  |-  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )  =  ( A  i^i  ( ( B  X.  _V )  i^i  ( C  X.  _V ) ) )
85, 6, 73eqtr4ri 2114 . 2  |-  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )  =  ( A  |`  ( B  i^i  C ) )
91, 3, 83eqtri 2107 1  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1285   _Vcvv 2612    i^i cin 2983    X. cxp 4399    |` cres 4403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4407  df-rel 4408  df-res 4413
This theorem is referenced by:  rescom  4695  resabs1  4699  resima2  4703  resmpt3  4718  resdisj  4813  rescnvcnv  4847  funimaexg  5051  fresin  5137  resdif  5223  pmresg  6363
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