Theorem relcnvtr 5263

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 4921	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
2		cnvss 4909	. . 3 |- ( ( R o. R ) C_ R -> `' ( R o. R ) C_ `' R )
3	1, 2	eqsstrrid 3275	. 2 |- ( ( R o. R ) C_ R -> ( `' R o. `' R ) C_ `' R )
4		cnvco 4921	. . . 4 |- `' ( `' R o. `' R ) = ( `' `' R o. `' `' R )
5		cnvss 4909	. . . 4 |- ( ( `' R o. `' R ) C_ `' R -> `' ( `' R o. `' R ) C_ `' `' R )
6		sseq1 3251	. . . . 5 |- ( `' ( `' R o. `' R ) = ( `' `' R o. `' `' R ) -> ( `' ( `' R o. `' R ) C_ `' `' R <-> ( `' `' R o. `' `' R ) C_ `' `' R ) )
7		dfrel2 5194	. . . . . . 7 |- ( Rel R <-> `' `' R = R )
8		coeq1 4893	. . . . . . . . . 10 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. `' `' R ) )
9		coeq2 4894	. . . . . . . . . 10 |- ( `' `' R = R -> ( R o. `' `' R ) = ( R o. R ) )
10	8, 9	eqtrd 2264	. . . . . . . . 9 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. R ) )
11		id 19	. . . . . . . . 9 |- ( `' `' R = R -> `' `' R = R )
12	10, 11	sseq12d 3259	. . . . . . . 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 1398   C_ wss 3201   `'ccnv 4730   o. ccom 4735   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740

This theorem is referenced by: (None)