Theorem relcnvtr 5066

Description: A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
relcnvtr	|- ( Rel R -> ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) )

Proof of Theorem relcnvtr

Step	Hyp	Ref	Expression
1		cnvco 4732	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
2		cnvss 4720	. . 3 |- ( ( R o. R ) C_ R -> `' ( R o. R ) C_ `' R )
3	1, 2	eqsstrrid 3149	. 2 |- ( ( R o. R ) C_ R -> ( `' R o. `' R ) C_ `' R )
4		cnvco 4732	. . . 4 |- `' ( `' R o. `' R ) = ( `' `' R o. `' `' R )
5		cnvss 4720	. . . 4 |- ( ( `' R o. `' R ) C_ `' R -> `' ( `' R o. `' R ) C_ `' `' R )
6		sseq1 3125	. . . . 5 |- ( `' ( `' R o. `' R ) = ( `' `' R o. `' `' R ) -> ( `' ( `' R o. `' R ) C_ `' `' R <-> ( `' `' R o. `' `' R ) C_ `' `' R ) )
7		dfrel2 4997	. . . . . . 7 |- ( Rel R <-> `' `' R = R )
8		coeq1 4704	. . . . . . . . . 10 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. `' `' R ) )
9		coeq2 4705	. . . . . . . . . 10 |- ( `' `' R = R -> ( R o. `' `' R ) = ( R o. R ) )
10	8, 9	eqtrd 2173	. . . . . . . . 9 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. R ) )
11		id 19	. . . . . . . . 9 |- ( `' `' R = R -> `' `' R = R )
12	10, 11	sseq12d 3133	. . . . . . . 8 |- ( `' `' R = R -> ( ( `' `' R o. `' `' R ) C_ `' `' R <-> ( R o. R ) C_ R ) )
13	12	biimpd 143	. . . . . . 7 |- ( `' `' R = R -> ( ( `' `' R o. `' `' R ) C_ `' `' R -> ( R o. R ) C_ R ) )
14	7, 13	sylbi 120	. . . . . 6 |- ( Rel R -> ( ( `' `' R o. `' `' R ) C_ `' `' R -> ( R o. R ) C_ R ) )
15	14	com12 30	. . . . 5 |- ( ( `' `' R o. `' `' R ) C_ `' `' R -> ( Rel R -> ( R o. R ) C_ R ) )
16	6, 15	syl6bi 162	. . . 4 |- ( `' ( `' R o. `' R ) = ( `' `' R o. `' `' R ) -> ( `' ( `' R o. `' R ) C_ `' `' R -> ( Rel R -> ( R o. R ) C_ R ) ) )
17	4, 5, 16	mpsyl 65	. . 3 |- ( ( `' R o. `' R ) C_ `' R -> ( Rel R -> ( R o. R ) C_ R ) )
18	17	com12 30	. 2 |- ( Rel R -> ( ( `' R o. `' R ) C_ `' R -> ( R o. R ) C_ R ) )
19	3, 18	impbid2 142	1 |- ( Rel R -> ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   C_ wss 3076   `'ccnv 4546   o. ccom 4551   Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556

This theorem is referenced by: (None)