Theorem cocnvcnv1 5212

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 5155	. . 3 |- `' `' A = ( A |` _V )
2	1	coeq1i 4855	. 2 |- ( `' `' A o. B ) = ( ( A |` _V ) o. B )
3		ssv 3223	. . 3 |- ran B C_ _V
4		cores 5205	. . 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 2228	1 |- ( `' `' A o. B ) = ( A o. B )

Syntax hints:   = wceq 1373   _Vcvv 2776   C_ wss 3174   `'ccnv 4692   ran crn 4694   |` cres 4695   o. ccom 4697

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705

This theorem is referenced by:  cores2  5214  coires1  5219  cofunex2g  6218