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Theorem cocnvcnv1 5308
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 5251 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21coeq1i 4956 . 2  |-  ( `' `' A  o.  B
)  =  ( ( A  |`  _V )  o.  B )
3 ssv 3427 . . 3  |-  ran  B  C_ 
_V
4 cores 5300 . . 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 2450 1  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3022    C_ wss 3379   `'ccnv 4795   ran crn 4797    |` cres 4798    o. ccom 4800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808
This theorem is referenced by:  cores2  5310  coires1  5315  cofunex2g  6716  mvdco  17029  deg1val  22987  trlcocnv  34199  trclubgNEW  36138  cnvtrrel  36175  trrelsuperrel2dg  36176
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