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Theorem cocnvcnv2 5200
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 5143 . . 3  |-  `' `' B  =  ( B  |` 
_V )
21coeq2i 4860 . 2  |-  ( A  o.  `' `' B
)  =  ( A  o.  ( B  |`  _V ) )
3 resco 5193 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  ( B  |`  _V ) )
4 relco 5187 . . 3  |-  Rel  ( A  o.  B )
5 dfrel3 5147 . . 3  |-  ( Rel  ( A  o.  B
)  <->  ( ( A  o.  B )  |`  _V )  =  ( A  o.  B )
)
64, 5mpbi 199 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  B )
72, 3, 63eqtr2i 2322 1  |-  ( A  o.  `' `' B
)  =  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801   `'ccnv 4704    |` cres 4707    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  dfdm2  5220  cofunex2g  5756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-res 4717
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