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Theorem cocnvcnv1 5459
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 5402 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21coeq1i 5110 . 2  |-  ( `' `' A  o.  B
)  =  ( ( A  |`  _V )  o.  B )
3 ssv 3487 . . 3  |-  ran  B  C_ 
_V
4 cores 5452 . . 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 2483 1  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3078    C_ wss 3439   `'ccnv 4950   ran crn 4952    |` cres 4953    o. ccom 4955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963
This theorem is referenced by:  cores2  5461  coires1  5466  cofunex2g  6655  mvdco  16073  deg1val  21703  deg1valOLD  21704  trlcocnv  34722
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