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Theorem resres 4984
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 4717 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  B )  i^i  ( C  X.  _V ) )
2 df-res 4717 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
32ineq1i 3379 . 2  |-  ( ( A  |`  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )
4 xpindir 4836 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
54ineq2i 3380 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
6 df-res 4717 . . 3  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
7 inass 3392 . . 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 2327 . 2  |-  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )  =  ( A  |`  ( B  i^i  C ) )
91, 3, 83eqtri 2320 1  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    i^i cin 3164    X. cxp 4703    |` cres 4707
This theorem is referenced by:  rescom  4996  resabs1  5000  resima2  5004  resmpt3  5017  resdisj  5121  rescnvcnv  5151  fresin  5426  resdif  5510  curry1  6226  curry2  6229  pmresg  6811  gruima  8440  rlimres  12048  lo1res  12049  rlimresb  12055  lo1eq  12058  rlimeq  12059  setsid  13203  sscres  13716  gsumzres  15210  txkgen  17362  tsmsres  17842  ressxms  18087  ressms  18088  dvres  19277  dvres3a  19280  cpnres  19302  dvmptres3  19321  rlimcnp2  20277  df1stres  23258  df2ndres  23259  indf1ofs  23624  wfrlem4  24330  frrlem4  24355  domrancur1b  25303  domrancur1c  25305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712  df-res 4717
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