Theorem cocnvcnv2 4832

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 4774	. . 3 |- `' `' B = ( B |` _V )
2	1	coeq2i 4496	. 2 |- ( A o. `' `' B ) = ( A o. ( B |` _V ) )
3		resco 4825	. 2 |- ( ( A o. B ) |` _V ) = ( A o. ( B |` _V ) )
4		relco 4819	. . 3 |- Rel ( A o. B )
5		dfrel3 4778	. . 3 |- ( Rel ( A o. B ) <-> ( ( A o. B ) |` _V ) = ( A o. B ) )
6	4, 5	mpbi 133	. 2 |- ( ( A o. B ) |` _V ) = ( A o. B )
7	2, 3, 6	3eqtr2i 2066	1 |- ( A o. `' `' B ) = ( A o. B )

Syntax hints:   = wceq 1243   _Vcvv 2557   `'ccnv 4344   |` cres 4347   o. ccom 4349   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-res 4357

This theorem is referenced by:  dfdm2  4852  cofunex2g  5739