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Theorem relcnvtr 5510
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 5177 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
2 cnvss 5164 . . 3  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
31, 2syl5eqssr 3534 . 2  |-  ( ( R  o.  R ) 
C_  R  ->  ( `' R  o.  `' R )  C_  `' R )
4 cnvco 5177 . . . 4  |-  `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )
5 cnvss 5164 . . . 4  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  `' ( `' R  o.  `' R
)  C_  `' `' R )
6 sseq1 3510 . . . . 5  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  <->  ( `' `' R  o.  `' `' R )  C_  `' `' R ) )
7 dfrel2 5441 . . . . . . 7  |-  ( Rel 
R  <->  `' `' R  =  R
)
8 coeq1 5149 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  `' `' R ) )
9 coeq2 5150 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( R  o.  `' `' R )  =  ( R  o.  R ) )
108, 9eqtrd 2495 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  R ) )
11 id 22 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  `' `' R  =  R
)
1210, 11sseq12d 3518 . . . . . . . 8  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  <->  ( R  o.  R )  C_  R
) )
1312biimpd 207 . . . . . . 7  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
147, 13sylbi 195 . . . . . 6  |-  ( Rel 
R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
1514com12 31 . . . . 5  |-  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) )
166, 15syl6bi 228 . . . 4  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) ) )
174, 5, 16mpsyl 63 . . 3  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  ( Rel  R  ->  ( R  o.  R
)  C_  R )
)
1817com12 31 . 2  |-  ( Rel 
R  ->  ( ( `' R  o.  `' R )  C_  `' R  ->  ( R  o.  R )  C_  R
) )
193, 18impbid2 204 1  |-  ( Rel 
R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    C_ wss 3461   `'ccnv 4987    o. ccom 4992   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997
This theorem is referenced by: (None)
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