Theorem cocnvcnv2 5243

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 5185	. . 3 |- `' `' B = ( B |` _V )
2	1	coeq2i 4885	. 2 |- ( A o. `' `' B ) = ( A o. ( B |` _V ) )
3		resco 5236	. 2 |- ( ( A o. B ) |` _V ) = ( A o. ( B |` _V ) )
4		relco 5230	. . 3 |- Rel ( A o. B )
5		dfrel3 5189	. . 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 2256	1 |- ( A o. `' `' B ) = ( A o. B )

Syntax hints:   = wceq 1395   _Vcvv 2799   `'ccnv 4719   |` cres 4722   o. ccom 4724   Rel wrel 4725

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-res 4732

This theorem is referenced by:  dfdm2  5266  cofunex2g  6264