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Theorem cnvtsr 14581
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 14580 . . 3  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
2 cnvps 14571 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
31, 2syl 16 . 2  |-  ( R  e.  TosetRel  ->  `' R  e.  PosetRel )
4 eqid 2387 . . . . 5  |-  dom  R  =  dom  R
54istsr 14576 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
65simprbi 451 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) )
74psrn 14568 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
81, 7syl 16 . . . 4  |-  ( R  e.  TosetRel  ->  dom  R  =  ran  R )
98, 8xpeq12d 4843 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  =  ( ran  R  X.  ran  R ) )
10 psrel 14562 . . . . . . 7  |-  ( R  e.  PosetRel  ->  Rel  R )
111, 10syl 16 . . . . . 6  |-  ( R  e.  TosetRel  ->  Rel  R )
12 dfrel2 5261 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
1311, 12sylib 189 . . . . 5  |-  ( R  e.  TosetRel  ->  `' `' R  =  R )
1413uneq2d 3444 . . . 4  |-  ( R  e.  TosetRel  ->  ( `' R  u.  `' `' R )  =  ( `' R  u.  R
) )
15 uncom 3434 . . . 4  |-  ( `' R  u.  R )  =  ( R  u.  `' R )
1614, 15syl6req 2436 . . 3  |-  ( R  e.  TosetRel  ->  ( R  u.  `' R )  =  ( `' R  u.  `' `' R ) )
176, 9, 163sstr3d 3333 . 2  |-  ( R  e.  TosetRel  ->  ( ran  R  X.  ran  R )  C_  ( `' R  u.  `' `' R ) )
18 df-rn 4829 . . 3  |-  ran  R  =  dom  `' R
1918istsr 14576 . 2  |-  ( `' R  e.  TosetRel  <->  ( `' R  e.  PosetRel  /\  ( ran  R  X.  ran  R
)  C_  ( `' R  u.  `' `' R ) ) )
203, 17, 19sylanbrc 646 1  |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    u. cun 3261    C_ wss 3263    X. cxp 4816   `'ccnv 4817   dom cdm 4818   ran crn 4819   Rel wrel 4823   PosetRelcps 14551    TosetRel ctsr 14552
This theorem is referenced by:  ordtbas2  17177  ordtrest2  17190  cnvordtrestixx  24115
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ps 14556  df-tsr 14557
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