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Theorem cnvtsr 16174
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 16173 . . 3  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
2 cnvps 16164 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
31, 2syl 17 . 2  |-  ( R  e.  TosetRel  ->  `' R  e.  PosetRel )
4 eqid 2402 . . . . 5  |-  dom  R  =  dom  R
54istsr 16169 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
65simprbi 462 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) )
74psrn 16161 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
81, 7syl 17 . . . 4  |-  ( R  e.  TosetRel  ->  dom  R  =  ran  R )
98sqxpeqd 4848 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  =  ( ran  R  X.  ran  R ) )
10 psrel 16155 . . . . . . 7  |-  ( R  e.  PosetRel  ->  Rel  R )
111, 10syl 17 . . . . . 6  |-  ( R  e.  TosetRel  ->  Rel  R )
12 dfrel2 5273 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
1311, 12sylib 196 . . . . 5  |-  ( R  e.  TosetRel  ->  `' `' R  =  R )
1413uneq2d 3596 . . . 4  |-  ( R  e.  TosetRel  ->  ( `' R  u.  `' `' R )  =  ( `' R  u.  R
) )
15 uncom 3586 . . . 4  |-  ( `' R  u.  R )  =  ( R  u.  `' R )
1614, 15syl6req 2460 . . 3  |-  ( R  e.  TosetRel  ->  ( R  u.  `' R )  =  ( `' R  u.  `' `' R ) )
176, 9, 163sstr3d 3483 . 2  |-  ( R  e.  TosetRel  ->  ( ran  R  X.  ran  R )  C_  ( `' R  u.  `' `' R ) )
18 df-rn 4833 . . 3  |-  ran  R  =  dom  `' R
1918istsr 16169 . 2  |-  ( `' R  e.  TosetRel  <->  ( `' R  e.  PosetRel  /\  ( ran  R  X.  ran  R
)  C_  ( `' R  u.  `' `' R ) ) )
203, 17, 19sylanbrc 662 1  |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    u. cun 3411    C_ wss 3413    X. cxp 4820   `'ccnv 4821   dom cdm 4822   ran crn 4823   Rel wrel 4827   PosetRelcps 16150    TosetRel ctsr 16151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ps 16152  df-tsr 16153
This theorem is referenced by:  ordtbas2  19983  ordtrest2  19996  cnvordtrestixx  28334
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