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Theorem cnvtsr 18633
Description: The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
cnvtsr (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )

Proof of Theorem cnvtsr
StepHypRef Expression
1 tsrps 18632 . . 3 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2 cnvps 18623 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . 2 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
4 eqid 2737 . . . . 5 dom 𝑅 = dom 𝑅
54istsr 18628 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
65simprbi 496 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅))
74psrn 18620 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
81, 7syl 17 . . . 4 (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
98sqxpeqd 5717 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10 psrel 18614 . . . . . . 7 (𝑅 ∈ PosetRel → Rel 𝑅)
111, 10syl 17 . . . . . 6 (𝑅 ∈ TosetRel → Rel 𝑅)
12 dfrel2 6209 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
1311, 12sylib 218 . . . . 5 (𝑅 ∈ TosetRel → 𝑅 = 𝑅)
1413uneq2d 4168 . . . 4 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
15 uncom 4158 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1614, 15eqtr2di 2794 . . 3 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
176, 9, 163sstr3d 4038 . 2 (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅))
18 df-rn 5696 . . 3 ran 𝑅 = dom 𝑅
1918istsr 18628 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅)))
203, 17, 19sylanbrc 583 1 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cun 3949  wss 3951   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  Rel wrel 5690  PosetRelcps 18609   TosetRel ctsr 18610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ps 18611  df-tsr 18612
This theorem is referenced by:  ordtbas2  23199  ordtrest2  23212  cnvordtrestixx  33912
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