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Theorem resres 4725
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 4450 . 2  |-  ( ( A  |`  B )  |`  C )  =  ( ( A  |`  B )  i^i  ( C  X.  _V ) )
2 df-res 4450 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
32ineq1i 3197 . 2  |-  ( ( A  |`  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )
4 xpindir 4572 . . . 4  |-  ( ( B  i^i  C )  X.  _V )  =  ( ( B  X.  _V )  i^i  ( C  X.  _V ) )
54ineq2i 3198 . . 3  |-  ( A  i^i  ( ( B  i^i  C )  X. 
_V ) )  =  ( A  i^i  (
( B  X.  _V )  i^i  ( C  X.  _V ) ) )
6 df-res 4450 . . 3  |-  ( A  |`  ( B  i^i  C
) )  =  ( A  i^i  ( ( B  i^i  C )  X.  _V ) )
7 inass 3210 . . 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 2119 . 2  |-  ( ( A  i^i  ( B  X.  _V ) )  i^i  ( C  X.  _V ) )  =  ( A  |`  ( B  i^i  C ) )
91, 3, 83eqtri 2112 1  |-  ( ( A  |`  B )  |`  C )  =  ( A  |`  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1289   _Vcvv 2619    i^i cin 2998    X. cxp 4436    |` cres 4440
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445  df-res 4450
This theorem is referenced by:  rescom  4738  resabs1  4742  resima2  4746  resmpt3  4761  resdisj  4859  rescnvcnv  4893  funimaexg  5098  fresin  5189  resdif  5275  pmresg  6433  setsslid  11544
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