Theorem cocnvcnv1 5131

Description: A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)

Assertion

Ref	Expression
cocnvcnv1	|- ( `' `' A o. B ) = ( A o. B )

Proof of Theorem cocnvcnv1

Step	Hyp	Ref	Expression
1		cnvcnv2 5074	. . 3 |- `' `' A = ( A |` _V )
2	1	coeq1i 4779	. 2 |- ( `' `' A o. B ) = ( ( A |` _V ) o. B )
3		ssv 3175	. . 3 |- ran B C_ _V
4		cores 5124	. . 3 |- ( ran B C_ _V -> ( ( A |` _V ) o. B ) = ( A o. B ) )
5	3, 4	ax-mp 5	. 2 |- ( ( A |` _V ) o. B ) = ( A o. B )
6	2, 5	eqtri 2196	1 |- ( `' `' A o. B ) = ( A o. B )

Syntax hints:   = wceq 1353   _Vcvv 2735   C_ wss 3127   `'ccnv 4619   ran crn 4621   |` cres 4622   o. ccom 4624

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632

This theorem is referenced by:  cores2  5133  coires1  5138  cofunex2g  6101