Theorem cocnvcnv2 4937

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 4879	. . 3 |- `' `' B = ( B |` _V )
2	1	coeq2i 4592	. 2 |- ( A o. `' `' B ) = ( A o. ( B |` _V ) )
3		resco 4930	. 2 |- ( ( A o. B ) |` _V ) = ( A o. ( B |` _V ) )
4		relco 4924	. . 3 |- Rel ( A o. B )
5		dfrel3 4883	. . 3 |- ( Rel ( A o. B ) <-> ( ( A o. B ) |` _V ) = ( A o. B ) )
6	4, 5	mpbi 143	. 2 |- ( ( A o. B ) |` _V ) = ( A o. B )
7	2, 3, 6	3eqtr2i 2114	1 |- ( A o. `' `' B ) = ( A o. B )

Syntax hints:   = wceq 1289   _Vcvv 2619   `'ccnv 4435   |` cres 4438   o. ccom 4440   Rel wrel 4441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-res 4448

This theorem is referenced by:  dfdm2  4960  cofunex2g  5875