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Theorem cocnvcnv2 5184
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 5127 . . 3  |-  `' `' B  =  ( B  |` 
_V )
21coeq2i 4844 . 2  |-  ( A  o.  `' `' B
)  =  ( A  o.  ( B  |`  _V ) )
3 resco 5177 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  ( B  |`  _V ) )
4 relco 5171 . . 3  |-  Rel  ( A  o.  B )
5 dfrel3 5131 . . 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 2309 1  |-  ( A  o.  `' `' B
)  =  ( A  o.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788   `'ccnv 4688    |` cres 4691    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  dfdm2  5204  cofunex2g  5740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701
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