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Theorem rescnvcnv 5317
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 5308 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21reseq1i 5120 . 2  |-  ( `' `' A  |`  B )  =  ( ( A  |`  _V )  |`  B )
3 resres 5136 . 2  |-  ( ( A  |`  _V )  |`  B )  =  ( A  |`  ( _V  i^i  B ) )
4 ssv 3464 . . . 4  |-  B  C_  _V
5 sseqin2 3663 . . . 4  |-  ( B 
C_  _V  <->  ( _V  i^i  B )  =  B )
64, 5mpbi 213 . . 3  |-  ( _V 
i^i  B )  =  B
76reseq2i 5121 . 2  |-  ( A  |`  ( _V  i^i  B
) )  =  ( A  |`  B )
82, 3, 73eqtri 2488 1  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   _Vcvv 3057    i^i cin 3415    C_ wss 3416   `'ccnv 4852    |` cres 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-res 4865
This theorem is referenced by:  cnvcnvres  5318  imacnvcnv  5319  resdm2  5344  resdmres  5345  coires1  5372  f1oresrab  6079
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