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Theorem cnvtsr 18646
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 18645 . . 3 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2 cnvps 18636 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . 2 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
4 eqid 2735 . . . . 5 dom 𝑅 = dom 𝑅
54istsr 18641 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
65simprbi 496 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅))
74psrn 18633 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
81, 7syl 17 . . . 4 (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
98sqxpeqd 5721 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10 psrel 18627 . . . . . . 7 (𝑅 ∈ PosetRel → Rel 𝑅)
111, 10syl 17 . . . . . 6 (𝑅 ∈ TosetRel → Rel 𝑅)
12 dfrel2 6211 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
1311, 12sylib 218 . . . . 5 (𝑅 ∈ TosetRel → 𝑅 = 𝑅)
1413uneq2d 4178 . . . 4 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
15 uncom 4168 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1614, 15eqtr2di 2792 . . 3 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
176, 9, 163sstr3d 4042 . 2 (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅))
18 df-rn 5700 . . 3 ran 𝑅 = dom 𝑅
1918istsr 18641 . 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 1537  wcel 2106  cun 3961  wss 3963   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  Rel wrel 5694  PosetRelcps 18622   TosetRel ctsr 18623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ps 18624  df-tsr 18625
This theorem is referenced by:  ordtbas2  23215  ordtrest2  23228  cnvordtrestixx  33874
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