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Theorem rescnvcnv 5135
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 5127 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21reseq1i 4951 . 2  |-  ( `' `' A  |`  B )  =  ( ( A  |`  _V )  |`  B )
3 resres 4968 . 2  |-  ( ( A  |`  _V )  |`  B )  =  ( A  |`  ( _V  i^i  B ) )
4 ssv 3198 . . . 4  |-  B  C_  _V
5 sseqin2 3388 . . . 4  |-  ( B 
C_  _V  <->  ( _V  i^i  B )  =  B )
64, 5mpbi 199 . . 3  |-  ( _V 
i^i  B )  =  B
76reseq2i 4952 . 2  |-  ( A  |`  ( _V  i^i  B
) )  =  ( A  |`  B )
82, 3, 73eqtri 2307 1  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151    C_ wss 3152   `'ccnv 4688    |` cres 4691
This theorem is referenced by:  cnvcnvres  5136  imacnvcnv  5137  resdm2  5163  resdmres  5164  coires1  5190  domrancur1c  25202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-res 4701
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