Theorem cocnvcnv1 5296

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 5239	. . 3 |- `' `' A = ( A |` _V )
2	1	coeq1i 4937	. 2 |- ( `' `' A o. B ) = ( ( A |` _V ) o. B )
3		ssv 3270	. . 3 |- ran B C_ _V
4		cores 5289	. . 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 2259	1 |- ( `' `' A o. B ) = ( A o. B )

Syntax hints:   = wceq 1402   _Vcvv 2821   C_ wss 3220   `'ccnv 4771   ran crn 4773   |` cres 4774   o. ccom 4776

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784

This theorem is referenced by:  cores2  5298  coires1  5303  cofunex2g  6333