Theorem relcnvtr 5163

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 4827	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
2		cnvss 4815	. . 3 |- ( ( R o. R ) C_ R -> `' ( R o. R ) C_ `' R )
3	1, 2	eqsstrrid 3217	. 2 |- ( ( R o. R ) C_ R -> ( `' R o. `' R ) C_ `' R )
4		cnvco 4827	. . . 4 |- `' ( `' R o. `' R ) = ( `' `' R o. `' `' R )
5		cnvss 4815	. . . 4 |- ( ( `' R o. `' R ) C_ `' R -> `' ( `' R o. `' R ) C_ `' `' R )
6		sseq1 3193	. . . . 5 |- ( `' ( `' R o. `' R ) = ( `' `' R o. `' `' R ) -> ( `' ( `' R o. `' R ) C_ `' `' R <-> ( `' `' R o. `' `' R ) C_ `' `' R ) )
7		dfrel2 5094	. . . . . . 7 |- ( Rel R <-> `' `' R = R )
8		coeq1 4799	. . . . . . . . . 10 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. `' `' R ) )
9		coeq2 4800	. . . . . . . . . 10 |- ( `' `' R = R -> ( R o. `' `' R ) = ( R o. R ) )
10	8, 9	eqtrd 2222	. . . . . . . . 9 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. R ) )
11		id 19	. . . . . . . . 9 |- ( `' `' R = R -> `' `' R = R )
12	10, 11	sseq12d 3201	. . . . . . . 8 |- ( `' `' R = R -> ( ( `' `' R o. `' `' R ) C_ `' `' R <-> ( R o. R ) C_ R ) )
13	12	biimpd 144	. . . . . . 7 |- ( `' `' R = R -> ( ( `' `' R o. `' `' R ) C_ `' `' R -> ( R o. R ) C_ R ) )
14	7, 13	sylbi 121	. . . . . 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	biimtrdi 163	. . . 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 143	1 |- ( Rel R -> ( ( R o. R ) C_ R <-> ( `' R o. `' R ) C_ `' R ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   C_ wss 3144   `'ccnv 4640   o. ccom 4645   Rel wrel 4646

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650

This theorem is referenced by: (None)