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Theorem cocnvcnv1 5524
Description: A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cocnvcnv1  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )

Proof of Theorem cocnvcnv1
StepHypRef Expression
1 cnvcnv2 5466 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21coeq1i 5168 . 2  |-  ( `' `' A  o.  B
)  =  ( ( A  |`  _V )  o.  B )
3 ssv 3529 . . 3  |-  ran  B  C_ 
_V
4 cores 5516 . . 3  |-  ( ran 
B  C_  _V  ->  ( ( A  |`  _V )  o.  B )  =  ( A  o.  B ) )
53, 4ax-mp 5 . 2  |-  ( ( A  |`  _V )  o.  B )  =  ( A  o.  B )
62, 5eqtri 2496 1  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3118    C_ wss 3481   `'ccnv 5004   ran crn 5006    |` cres 5007    o. ccom 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017
This theorem is referenced by:  cores2  5526  coires1  5531  cofunex2g  6760  mvdco  16343  deg1val  22364  deg1valOLD  22365  trlcocnv  35917  cnvtrrel  37193
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