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Theorem cocnvcnv2 5158
Description: A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cocnvcnv2  |-  ( A  o.  `' `' B
)  =  ( A  o.  B )

Proof of Theorem cocnvcnv2
StepHypRef Expression
1 cnvcnv2 5100 . . 3  |-  `' `' B  =  ( B  |` 
_V )
21coeq2i 4805 . 2  |-  ( A  o.  `' `' B
)  =  ( A  o.  ( B  |`  _V ) )
3 resco 5151 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  ( B  |`  _V ) )
4 relco 5145 . . 3  |-  Rel  ( A  o.  B )
5 dfrel3 5104 . . 3  |-  ( Rel  ( A  o.  B
)  <->  ( ( A  o.  B )  |`  _V )  =  ( A  o.  B )
)
64, 5mpbi 145 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  B )
72, 3, 63eqtr2i 2216 1  |-  ( A  o.  `' `' B
)  =  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1364   _Vcvv 2752   `'ccnv 4643    |` cres 4646    o. ccom 4648   Rel wrel 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-res 4656
This theorem is referenced by:  dfdm2  5181  cofunex2g  6136
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