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Theorem cnvtsr 14646
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 14645 . . 3  |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
2 cnvps 14636 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
31, 2syl 16 . 2  |-  ( R  e.  TosetRel  ->  `' R  e.  PosetRel )
4 eqid 2435 . . . . 5  |-  dom  R  =  dom  R
54istsr 14641 . . . 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 14633 . . . . 5  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
81, 7syl 16 . . . 4  |-  ( R  e.  TosetRel  ->  dom  R  =  ran  R )
98, 8xpeq12d 4895 . . 3  |-  ( R  e.  TosetRel  ->  ( dom  R  X.  dom  R )  =  ( ran  R  X.  ran  R ) )
10 psrel 14627 . . . . . . 7  |-  ( R  e.  PosetRel  ->  Rel  R )
111, 10syl 16 . . . . . 6  |-  ( R  e.  TosetRel  ->  Rel  R )
12 dfrel2 5313 . . . . . 6  |-  ( Rel 
R  <->  `' `' R  =  R
)
1311, 12sylib 189 . . . . 5  |-  ( R  e.  TosetRel  ->  `' `' R  =  R )
1413uneq2d 3493 . . . 4  |-  ( R  e.  TosetRel  ->  ( `' R  u.  `' `' R )  =  ( `' R  u.  R
) )
15 uncom 3483 . . . 4  |-  ( `' R  u.  R )  =  ( R  u.  `' R )
1614, 15syl6req 2484 . . 3  |-  ( R  e.  TosetRel  ->  ( R  u.  `' R )  =  ( `' R  u.  `' `' R ) )
176, 9, 163sstr3d 3382 . 2  |-  ( R  e.  TosetRel  ->  ( ran  R  X.  ran  R )  C_  ( `' R  u.  `' `' R ) )
18 df-rn 4881 . . 3  |-  ran  R  =  dom  `' R
1918istsr 14641 . 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 1652    e. wcel 1725    u. cun 3310    C_ wss 3312    X. cxp 4868   `'ccnv 4869   dom cdm 4870   ran crn 4871   Rel wrel 4875   PosetRelcps 14616    TosetRel ctsr 14617
This theorem is referenced by:  ordtbas2  17247  ordtrest2  17260  cnvordtrestixx  24303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ps 14621  df-tsr 14622
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