Theorem cocnvcnv2 5276

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

Step	Hyp	Ref	Expression
1		cnvcnv2 5218	. . 3 |- `' `' B = ( B |` _V )
2	1	coeq2i 4917	. 2 |- ( A o. `' `' B ) = ( A o. ( B |` _V ) )
3		resco 5269	. 2 |- ( ( A o. B ) |` _V ) = ( A o. ( B |` _V ) )
4		relco 5263	. . 3 |- Rel ( A o. B )
5		dfrel3 5222	. . 3 |- ( Rel ( A o. B ) <-> ( ( A o. B ) |` _V ) = ( A o. B ) )
6	4, 5	mpbi 145	. 2 |- ( ( A o. B ) |` _V ) = ( A o. B )
7	2, 3, 6	3eqtr2i 2261	1 |- ( A o. `' `' B ) = ( A o. B )

Syntax hints:   = wceq 1398   _Vcvv 2815   `'ccnv 4750   |` cres 4753   o. ccom 4755   Rel wrel 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-res 4763

This theorem is referenced by:  dfdm2  5299  cofunex2g  6305