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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
StepHypRef 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
)
31, 2eqsstrrid 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 ) )
108, 9eqtrd 2222 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  R ) )
11 id 19 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  `' `' R  =  R
)
1210, 11sseq12d 3201 . . . . . . . 8  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  <->  ( R  o.  R )  C_  R
) )
1312biimpd 144 . . . . . . 7  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
147, 13sylbi 121 . . . . . 6  |-  ( Rel 
R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
1514com12 30 . . . . 5  |-  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) )
166, 15biimtrdi 163 . . . 4  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) ) )
174, 5, 16mpsyl 65 . . 3  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  ( Rel  R  ->  ( R  o.  R
)  C_  R )
)
1817com12 30 . 2  |-  ( Rel 
R  ->  ( ( `' R  o.  `' R )  C_  `' R  ->  ( R  o.  R )  C_  R
) )
193, 18impbid2 143 1  |-  ( Rel 
R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
Colors of variables: wff set class
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)
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