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Theorem rescnvcnv 5106
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 5097 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21reseq1i 4918 . 2  |-  ( `' `' A  |`  B )  =  ( ( A  |`  _V )  |`  B )
3 resres 4934 . 2  |-  ( ( A  |`  _V )  |`  B )  =  ( A  |`  ( _V  i^i  B ) )
4 ssv 3192 . . . 4  |-  B  C_  _V
5 sseqin2 3369 . . . 4  |-  ( B 
C_  _V  <->  ( _V  i^i  B )  =  B )
64, 5mpbi 145 . . 3  |-  ( _V 
i^i  B )  =  B
76reseq2i 4919 . 2  |-  ( A  |`  ( _V  i^i  B
) )  =  ( A  |`  B )
82, 3, 73eqtri 2214 1  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1364   _Vcvv 2752    i^i cin 3143    C_ wss 3144   `'ccnv 4640    |` cres 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-res 4653
This theorem is referenced by:  cnvcnvres  5107  imacnvcnv  5108  resdm2  5134  resdmres  5135  coires1  5161  cocnvres  5168  f1oresrab  5697
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