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Theorem relcnvtr 5322
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 4989 . . 3  |-  `' ( R  o.  R )  =  ( `' R  o.  `' R )
2 cnvss 4978 . . 3  |-  ( ( R  o.  R ) 
C_  R  ->  `' ( R  o.  R
)  C_  `' R
)
31, 2syl5eqssr 3329 . 2  |-  ( ( R  o.  R ) 
C_  R  ->  ( `' R  o.  `' R )  C_  `' R )
4 cnvco 4989 . . . 4  |-  `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )
5 cnvss 4978 . . . 4  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  `' ( `' R  o.  `' R
)  C_  `' `' R )
6 sseq1 3305 . . . . 5  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  <->  ( `' `' R  o.  `' `' R )  C_  `' `' R ) )
7 dfrel2 5254 . . . . . . 7  |-  ( Rel 
R  <->  `' `' R  =  R
)
8 coeq1 4963 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  `' `' R ) )
9 coeq2 4964 . . . . . . . . . 10  |-  ( `' `' R  =  R  ->  ( R  o.  `' `' R )  =  ( R  o.  R ) )
108, 9eqtrd 2412 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  ( `' `' R  o.  `' `' R )  =  ( R  o.  R ) )
11 id 20 . . . . . . . . 9  |-  ( `' `' R  =  R  ->  `' `' R  =  R
)
1210, 11sseq12d 3313 . . . . . . . 8  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  <->  ( R  o.  R )  C_  R
) )
1312biimpd 199 . . . . . . 7  |-  ( `' `' R  =  R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
147, 13sylbi 188 . . . . . 6  |-  ( Rel 
R  ->  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( R  o.  R )  C_  R ) )
1514com12 29 . . . . 5  |-  ( ( `' `' R  o.  `' `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) )
166, 15syl6bi 220 . . . 4  |-  ( `' ( `' R  o.  `' R )  =  ( `' `' R  o.  `' `' R )  ->  ( `' ( `' R  o.  `' R )  C_  `' `' R  ->  ( Rel 
R  ->  ( R  o.  R )  C_  R
) ) )
174, 5, 16mpsyl 61 . . 3  |-  ( ( `' R  o.  `' R )  C_  `' R  ->  ( Rel  R  ->  ( R  o.  R
)  C_  R )
)
1817com12 29 . 2  |-  ( Rel 
R  ->  ( ( `' R  o.  `' R )  C_  `' R  ->  ( R  o.  R )  C_  R
) )
193, 18impbid2 196 1  |-  ( Rel 
R  ->  ( ( R  o.  R )  C_  R  <->  ( `' R  o.  `' R )  C_  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    C_ wss 3256   `'ccnv 4810    o. ccom 4815   Rel wrel 4816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820
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