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Theorem rescnvcnv 4834
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 4825 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21reseq1i 4657 . 2  |-  ( `' `' A  |`  B )  =  ( ( A  |`  _V )  |`  B )
3 resres 4673 . 2  |-  ( ( A  |`  _V )  |`  B )  =  ( A  |`  ( _V  i^i  B ) )
4 ssv 3029 . . . 4  |-  B  C_  _V
5 sseqin2 3202 . . . 4  |-  ( B 
C_  _V  <->  ( _V  i^i  B )  =  B )
64, 5mpbi 143 . . 3  |-  ( _V 
i^i  B )  =  B
76reseq2i 4658 . 2  |-  ( A  |`  ( _V  i^i  B
) )  =  ( A  |`  B )
82, 3, 73eqtri 2107 1  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1285   _Vcvv 2610    i^i cin 2982    C_ wss 2983   `'ccnv 4391    |` cres 4394
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-rel 4399  df-cnv 4400  df-res 4404
This theorem is referenced by:  cnvcnvres  4835  imacnvcnv  4836  resdm2  4862  resdmres  4863  coires1  4889  f1oresrab  5382
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