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Theorem resres 5161
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 4892 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  B )  i^i  ( C  X.  _V ) )
2 df-res 4892 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
32ineq1i 3540 . 2  |-  ( ( A  |`  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )
4 xpindir 5011 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
54ineq2i 3541 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
6 df-res 4892 . . 3  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
7 inass 3553 . . 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 2469 . 2  |-  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )  =  ( A  |`  ( B  i^i  C ) )
91, 3, 83eqtri 2462 1  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    i^i cin 3321    X. cxp 4878    |` cres 4882
This theorem is referenced by:  rescom  5173  resabs1  5177  resima2  5181  resmpt3  5194  resdisj  5300  rescnvcnv  5334  fresin  5614  resdif  5698  curry1  6440  curry2  6443  pmresg  7043  gruima  8679  rlimres  12354  lo1res  12355  rlimresb  12361  lo1eq  12364  rlimeq  12365  setsid  13510  sscres  14025  gsumzres  15519  txkgen  17686  tsmsres  18175  ressxms  18557  ressms  18558  dvres  19800  dvres3a  19803  cpnres  19825  dvmptres3  19844  rlimcnp2  20807  df1stres  24093  df2ndres  24094  indf1ofs  24425  wfrlem4  25543  frrlem4  25587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887  df-res 4892
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