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Theorem rescnvcnv 4906
Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
rescnvcnv  |-  ( `' `' A  |`  B )  =  ( A  |`  B )

Proof of Theorem rescnvcnv
StepHypRef Expression
1 cnvcnv2 4897 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21reseq1i 4722 . 2  |-  ( `' `' A  |`  B )  =  ( ( A  |`  _V )  |`  B )
3 resres 4738 . 2  |-  ( ( A  |`  _V )  |`  B )  =  ( A  |`  ( _V  i^i  B ) )
4 ssv 3047 . . . 4  |-  B  C_  _V
5 sseqin2 3220 . . . 4  |-  ( B 
C_  _V  <->  ( _V  i^i  B )  =  B )
64, 5mpbi 144 . . 3  |-  ( _V 
i^i  B )  =  B
76reseq2i 4723 . 2  |-  ( A  |`  ( _V  i^i  B
) )  =  ( A  |`  B )
82, 3, 73eqtri 2113 1  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1290   _Vcvv 2620    i^i cin 2999    C_ wss 3000   `'ccnv 4450    |` cres 4453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4457  df-rel 4458  df-cnv 4459  df-res 4463
This theorem is referenced by:  cnvcnvres  4907  imacnvcnv  4908  resdm2  4934  resdmres  4935  coires1  4961  cocnvres  4968  f1oresrab  5477
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