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Theorem cnvtsr 14347
Description: The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
cnvtsr  |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )

Proof of Theorem cnvtsr
StepHypRef Expression
1 tsrps 14346 . . 3  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
2 cnvps 14337 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
31, 2syl 15 . 2  |-  ( R  e.  TosetRel  ->  `' R  e.  PosetRel )
4 eqid 2296 . . . . 5  |-  dom  R  =  dom  R
54istsr 14342 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
65simprbi 450 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) )
74psrn 14334 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
81, 7syl 15 . . . 4  |-  ( R  e.  TosetRel  ->  dom  R  =  ran  R )
98, 8xpeq12d 4730 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  =  ( ran  R  X.  ran  R ) )
10 psrel 14328 . . . . . . 7  |-  ( R  e.  PosetRel  ->  Rel  R )
111, 10syl 15 . . . . . 6  |-  ( R  e.  TosetRel  ->  Rel  R )
12 dfrel2 5140 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
1311, 12sylib 188 . . . . 5  |-  ( R  e.  TosetRel  ->  `' `' R  =  R )
1413uneq2d 3342 . . . 4  |-  ( R  e.  TosetRel  ->  ( `' R  u.  `' `' R )  =  ( `' R  u.  R
) )
15 uncom 3332 . . . 4  |-  ( `' R  u.  R )  =  ( R  u.  `' R )
1614, 15syl6req 2345 . . 3  |-  ( R  e.  TosetRel  ->  ( R  u.  `' R )  =  ( `' R  u.  `' `' R ) )
176, 9, 163sstr3d 3233 . 2  |-  ( R  e.  TosetRel  ->  ( ran  R  X.  ran  R )  C_  ( `' R  u.  `' `' R ) )
18 df-rn 4716 . . 3  |-  ran  R  =  dom  `' R
1918istsr 14342 . 2  |-  ( `' R  e.  TosetRel  <->  ( `' R  e.  PosetRel  /\  ( ran  R  X.  ran  R
)  C_  ( `' R  u.  `' `' R ) ) )
203, 17, 19sylanbrc 645 1  |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   Rel wrel 4710   PosetRelcps 14317    TosetRel ctsr 14318
This theorem is referenced by:  ordtbas2  16937  ordtrest2  16950  cnvordtrestixx  23312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ps 14322  df-tsr 14323
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