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Theorem cocnvcnv2 5502
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 5444 . . 3  |-  `' `' B  =  ( B  |` 
_V )
21coeq2i 5152 . 2  |-  ( A  o.  `' `' B
)  =  ( A  o.  ( B  |`  _V ) )
3 resco 5494 . 2  |-  ( ( A  o.  B )  |`  _V )  =  ( A  o.  ( B  |`  _V ) )
4 relco 5488 . . 3  |-  Rel  ( A  o.  B )
5 dfrel3 5448 . . 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 2489 1  |-  ( A  o.  `' `' B
)  =  ( A  o.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   _Vcvv 3106   `'ccnv 4987    |` cres 4990    o. ccom 4992   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-res 5000
This theorem is referenced by:  dfdm2  5522  cofunex2g  6738  cnvtrrel  38210
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