Theorem relcnvtr 4783

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 4463	. . 3 |- `'( R o. R) = ( `' R o. `' R)
2		cnvss 4451	. . 3 |- (( R o. R) C_ R -> `'( R o. R) C_ `' R)
3	1, 2	syl5eqssr 2984	. 2 |- (( R o. R) C_ R -> ( `' R o. `' R) C_ `' R)
4		cnvco 4463	. . . 4 |- `'( `' R o. `' R) = ( `' `' R o. `' `' R)
5		cnvss 4451	. . . 4 |- (( `' R o. `' R) C_ `' R -> `'( `' R o. `' R) C_ `' `' R)
6		sseq1 2960	. . . . 5 |- ( `'( `' R o. `' R) = ( `' `' R o. `' `' R) -> ( `'( `' R o. `' R) C_ `' `' R <-> ( `' `' R o. `' `' R) C_ `' `' R))
7		dfrel2 4714	. . . . . . 7 |- ( Rel R <-> `' `' R = R)
8		coeq1 4436	. . . . . . . . . 10 |- ( `' `' R = R -> ( `' `' R o. `' `' R) = ( R o. `' `' R))
9		coeq2 4437	. . . . . . . . . 10 |- ( `' `' R = R -> ( R o. `' `' R) = ( R o. R))
10	8, 9	eqtrd 2069	. . . . . . . . 9 |- ( `' `' R = R -> ( `' `' R o. `' `' R) = ( R o. R))
11		id 19	. . . . . . . . 9 |- ( `' `' R = R -> `' `' R = R)
12	10, 11	sseq12d 2968	. . . . . . . 8 |- ( `' `' R = R -> (( `' `' R o. `' `' R) C_ `' `' R <-> ( R o. R) C_ R))
13	12	biimpd 132	. . . . . . 7 |- ( `' `' R = R -> (( `' `' R o. `' `' R) C_ `' `' R -> ( R o. R) C_ R))
14	7, 13	sylbi 114	. . . . . 6 |- ( Rel R -> (( `' `' R o. `' `' R) C_ `' `' R -> ( R o. R) C_ R))
15	14	com12 27	. . . . 5 |- (( `' `' R o. `' `' R) C_ `' `' R -> ( Rel R -> ( R o. R) C_ R))
16	6, 15	syl6bi 152	. . . 4 |- ( `'( `' R o. `' R) = ( `' `' R o. `' `' R) -> ( `'( `' R o. `' R) C_ `' `' R -> ( Rel R -> ( R o. R) C_ R)))
17	4, 5, 16	mpsyl 59	. . 3 |- (( `' R o. `' R) C_ `' R -> ( Rel R -> ( R o. R) C_ R))
18	17	com12 27	. 2 |- ( Rel R -> (( `' R o. `' R) C_ `' R -> ( R o. R) C_ R))
19	3, 18	impbid2 131	1 |- ( Rel R -> (( R o. R) C_ R <-> ( `' R o. `' R) C_ `' R))

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1242   C_ wss 2911   `'ccnv 4287   o. ccom 4292   Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297

This theorem is referenced by: (None)