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Theorem relcnvtr 4950
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 4621 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
2 cnvss 4609 . . 3  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
31, 2syl5eqssr 3071 . 2  |-  ( ( R  o.  R ) 
C_  R  ->  ( `' R  o.  `' R )  C_  `' R )
4 cnvco 4621 . . . 4  |-  `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )
5 cnvss 4609 . . . 4  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  `' ( `' R  o.  `' R
)  C_  `' `' R )
6 sseq1 3047 . . . . 5  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  <->  ( `' `' R  o.  `' `' R )  C_  `' `' R ) )
7 dfrel2 4881 . . . . . . 7  |-  ( Rel 
R  <->  `' `' R  =  R
)
8 coeq1 4593 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  `' `' R ) )
9 coeq2 4594 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( R  o.  `' `' R )  =  ( R  o.  R ) )
108, 9eqtrd 2120 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  R ) )
11 id 19 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  `' `' R  =  R
)
1210, 11sseq12d 3055 . . . . . . . 8  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  <->  ( R  o.  R )  C_  R
) )
1312biimpd 142 . . . . . . 7  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
147, 13sylbi 119 . . . . . 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, 15syl6bi 161 . . . 4  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) ) )
174, 5, 16mpsyl 64 . . 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 141 1  |-  ( Rel 
R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    C_ wss 2999   `'ccnv 4437    o. ccom 4442   Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447
This theorem is referenced by: (None)
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