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Theorem relcnvtr 5192
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
StepHypRef Expression
1 cnvco 4865 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
2 cnvss 4854 . . 3  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
31, 2syl5eqssr 3223 . 2  |-  ( ( R  o.  R ) 
C_  R  ->  ( `' R  o.  `' R )  C_  `' R )
4 cnvco 4865 . . . 4  |-  `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )
5 cnvss 4854 . . . 4  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  `' ( `' R  o.  `' R
)  C_  `' `' R )
6 sseq1 3199 . . . . 5  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  <->  ( `' `' R  o.  `' `' R )  C_  `' `' R ) )
7 dfrel2 5124 . . . . . . 7  |-  ( Rel 
R  <->  `' `' R  =  R
)
8 coeq1 4841 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  `' `' R ) )
9 coeq2 4842 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( R  o.  `' `' R )  =  ( R  o.  R ) )
108, 9eqtrd 2315 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  R ) )
11 id 19 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  `' `' R  =  R
)
1210, 11sseq12d 3207 . . . . . . . 8  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  <->  ( R  o.  R )  C_  R
) )
1312biimpd 198 . . . . . . 7  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
147, 13sylbi 187 . . . . . 6  |-  ( Rel 
R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
1514com12 27 . . . . 5  |-  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) )
166, 15syl6bi 219 . . . 4  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) ) )
174, 5, 16mpsyl 59 . . 3  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  ( Rel  R  ->  ( R  o.  R
)  C_  R )
)
1817com12 27 . 2  |-  ( Rel 
R  ->  ( ( `' R  o.  `' R )  C_  `' R  ->  ( R  o.  R )  C_  R
) )
193, 18impbid2 195 1  |-  ( Rel 
R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C_ wss 3152   `'ccnv 4688    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  dupre1  25243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698
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