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Theorem cnvtsr 15705
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 15704 . . 3  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
2 cnvps 15695 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
31, 2syl 16 . 2  |-  ( R  e.  TosetRel  ->  `' R  e.  PosetRel )
4 eqid 2467 . . . . 5  |-  dom  R  =  dom  R
54istsr 15700 . . . 4  |-  ( R  e.  TosetRel 
<->  ( R  e.  PosetRel  /\  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) ) )
65simprbi 464 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  C_  ( R  u.  `' R ) )
74psrn 15692 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
81, 7syl 16 . . . 4  |-  ( R  e.  TosetRel  ->  dom  R  =  ran  R )
98, 8xpeq12d 5024 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  =  ( ran  R  X.  ran  R ) )
10 psrel 15686 . . . . . . 7  |-  ( R  e.  PosetRel  ->  Rel  R )
111, 10syl 16 . . . . . 6  |-  ( R  e.  TosetRel  ->  Rel  R )
12 dfrel2 5455 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
1311, 12sylib 196 . . . . 5  |-  ( R  e.  TosetRel  ->  `' `' R  =  R )
1413uneq2d 3658 . . . 4  |-  ( R  e.  TosetRel  ->  ( `' R  u.  `' `' R )  =  ( `' R  u.  R
) )
15 uncom 3648 . . . 4  |-  ( `' R  u.  R )  =  ( R  u.  `' R )
1614, 15syl6req 2525 . . 3  |-  ( R  e.  TosetRel  ->  ( R  u.  `' R )  =  ( `' R  u.  `' `' R ) )
176, 9, 163sstr3d 3546 . 2  |-  ( R  e.  TosetRel  ->  ( ran  R  X.  ran  R )  C_  ( `' R  u.  `' `' R ) )
18 df-rn 5010 . . 3  |-  ran  R  =  dom  `' R
1918istsr 15700 . 2  |-  ( `' R  e.  TosetRel  <->  ( `' R  e.  PosetRel  /\  ( ran  R  X.  ran  R
)  C_  ( `' R  u.  `' `' R ) ) )
203, 17, 19sylanbrc 664 1  |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   Rel wrel 5004   PosetRelcps 15681    TosetRel ctsr 15682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ps 15683  df-tsr 15684
This theorem is referenced by:  ordtbas2  19458  ordtrest2  19471  cnvordtrestixx  27531
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