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Theorem cnvtsr 17143
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 17142 . . 3 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2 cnvps 17133 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . 2 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
4 eqid 2621 . . . . 5 dom 𝑅 = dom 𝑅
54istsr 17138 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
65simprbi 480 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅))
74psrn 17130 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
81, 7syl 17 . . . 4 (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
98sqxpeqd 5101 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10 psrel 17124 . . . . . . 7 (𝑅 ∈ PosetRel → Rel 𝑅)
111, 10syl 17 . . . . . 6 (𝑅 ∈ TosetRel → Rel 𝑅)
12 dfrel2 5542 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
1311, 12sylib 208 . . . . 5 (𝑅 ∈ TosetRel → 𝑅 = 𝑅)
1413uneq2d 3745 . . . 4 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
15 uncom 3735 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1614, 15syl6req 2672 . . 3 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
176, 9, 163sstr3d 3626 . 2 (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅))
18 df-rn 5085 . . 3 ran 𝑅 = dom 𝑅
1918istsr 17138 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅)))
203, 17, 19sylanbrc 697 1 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cun 3553  wss 3555   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  Rel wrel 5079  PosetRelcps 17119   TosetRel ctsr 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ps 17121  df-tsr 17122
This theorem is referenced by:  ordtbas2  20905  ordtrest2  20918  cnvordtrestixx  29741
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