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Theorem cocnvcnv2 5517
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 5458 . . 3  |-  `' `' B  =  ( B  |` 
_V )
21coeq2i 5161 . 2  |-  ( A  o.  `' `' B
)  =  ( A  o.  ( B  |`  _V ) )
3 resco 5509 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  ( B  |`  _V ) )
4 relco 5503 . . 3  |-  Rel  ( A  o.  B )
5 dfrel3 5462 . . 3  |-  ( Rel  ( A  o.  B
)  <->  ( ( A  o.  B )  |`  _V )  =  ( A  o.  B )
)
64, 5mpbi 208 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  B )
72, 3, 63eqtr2i 2502 1  |-  ( A  o.  `' `' B
)  =  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113   `'ccnv 4998    |` cres 5001    o. ccom 5003   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-res 5011
This theorem is referenced by:  dfdm2  5537  cofunex2g  6746  cnvtrrel  36792
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