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Theorem resres 4968
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 4701 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  B )  i^i  ( C  X.  _V ) )
2 df-res 4701 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
32ineq1i 3366 . 2  |-  ( ( A  |`  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )
4 xpindir 4820 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
54ineq2i 3367 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
6 df-res 4701 . . 3  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
7 inass 3379 . . 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 2314 . 2  |-  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )  =  ( A  |`  ( B  i^i  C ) )
91, 3, 83eqtri 2307 1  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151    X. cxp 4687    |` cres 4691
This theorem is referenced by:  rescom  4980  resabs1  4984  resima2  4988  resmpt3  5001  resdisj  5105  rescnvcnv  5135  fresin  5410  resdif  5494  curry1  6210  curry2  6213  pmresg  6795  gruima  8424  rlimres  12032  lo1res  12033  rlimresb  12039  lo1eq  12042  rlimeq  12043  setsid  13187  sscres  13700  gsumzres  15194  txkgen  17346  tsmsres  17826  ressxms  18071  ressms  18072  dvres  19261  dvres3a  19264  cpnres  19286  dvmptres3  19305  rlimcnp2  20261  df1stres  23243  df2ndres  23244  indf1ofs  23609  wfrlem4  24259  frrlem4  24284  domrancur1b  25200  domrancur1c  25202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-res 4701
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