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Theorem cocnvcnv1 5372
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 5316 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21coeq1i 5024 . 2  |-  ( `' `' A  o.  B
)  =  ( ( A  |`  _V )  o.  B )
3 ssv 3360 . . 3  |-  ran  B  C_ 
_V
4 cores 5365 . . 3  |-  ( ran 
B  C_  _V  ->  ( ( A  |`  _V )  o.  B )  =  ( A  o.  B ) )
53, 4ax-mp 8 . 2  |-  ( ( A  |`  _V )  o.  B )  =  ( A  o.  B )
62, 5eqtri 2455 1  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   _Vcvv 2948    C_ wss 3312   `'ccnv 4869   ran crn 4871    |` cres 4872    o. ccom 4874
This theorem is referenced by:  cores2  5374  coires1  5379  cofunex2g  5952  deg1val  20011  mvdco  27356  trlcocnv  31454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882
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