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Theorem cocnvcnv1 4861
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 4804 . . 3  |-  `' `' A  =  ( A  |` 
_V )
21coeq1i 4523 . 2  |-  ( `' `' A  o.  B
)  =  ( ( A  |`  _V )  o.  B )
3 ssv 3020 . . 3  |-  ran  B  C_ 
_V
4 cores 4854 . . 3  |-  ( ran 
B  C_  _V  ->  ( ( A  |`  _V )  o.  B )  =  ( A  o.  B ) )
53, 4ax-mp 7 . 2  |-  ( ( A  |`  _V )  o.  B )  =  ( A  o.  B )
62, 5eqtri 2102 1  |-  ( `' `' A  o.  B
)  =  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1285   _Vcvv 2602    C_ wss 2974   `'ccnv 4370   ran crn 4372    |` cres 4373    o. ccom 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383
This theorem is referenced by:  cores2  4863  coires1  4868  cofunex2g  5770
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