Theorem relcnvtr 5256

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 4915	. . 3 |- `' ( R o. R ) = ( `' R o. `' R )
2		cnvss 4903	. . 3 |- ( ( R o. R ) C_ R -> `' ( R o. R ) C_ `' R )
3	1, 2	eqsstrrid 3274	. 2 |- ( ( R o. R ) C_ R -> ( `' R o. `' R ) C_ `' R )
4		cnvco 4915	. . . 4 |- `' ( `' R o. `' R ) = ( `' `' R o. `' `' R )
5		cnvss 4903	. . . 4 |- ( ( `' R o. `' R ) C_ `' R -> `' ( `' R o. `' R ) C_ `' `' R )
6		sseq1 3250	. . . . 5 |- ( `' ( `' R o. `' R ) = ( `' `' R o. `' `' R ) -> ( `' ( `' R o. `' R ) C_ `' `' R <-> ( `' `' R o. `' `' R ) C_ `' `' R ) )
7		dfrel2 5187	. . . . . . 7 |- ( Rel R <-> `' `' R = R )
8		coeq1 4887	. . . . . . . . . 10 |- ( `' `' R = R -> ( `' `' R o. `' `' R ) = ( R o. `' `' R ) )
9		coeq2 4888	. . . . . . . . . 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 3258	. . . . . . . 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 1397   C_ wss 3200   `'ccnv 4724   o. ccom 4729   Rel wrel 4730

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734

This theorem is referenced by: (None)