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Theorem cnvtsr 17444
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 17443 . . 3 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2 cnvps 17434 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . 2 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
4 eqid 2761 . . . . 5 dom 𝑅 = dom 𝑅
54istsr 17439 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅)))
65simprbi 483 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅𝑅))
74psrn 17431 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
81, 7syl 17 . . . 4 (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
98sqxpeqd 5299 . . 3 (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10 psrel 17425 . . . . . . 7 (𝑅 ∈ PosetRel → Rel 𝑅)
111, 10syl 17 . . . . . 6 (𝑅 ∈ TosetRel → Rel 𝑅)
12 dfrel2 5742 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
1311, 12sylib 208 . . . . 5 (𝑅 ∈ TosetRel → 𝑅 = 𝑅)
1413uneq2d 3911 . . . 4 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
15 uncom 3901 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1614, 15syl6req 2812 . . 3 (𝑅 ∈ TosetRel → (𝑅𝑅) = (𝑅𝑅))
176, 9, 163sstr3d 3789 . 2 (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅))
18 df-rn 5278 . . 3 ran 𝑅 = dom 𝑅
1918istsr 17439 . 2 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (𝑅𝑅)))
203, 17, 19sylanbrc 701 1 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2140  cun 3714  wss 3716   × cxp 5265  ccnv 5266  dom cdm 5267  ran crn 5268  Rel wrel 5272  PosetRelcps 17420   TosetRel ctsr 17421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ps 17422  df-tsr 17423
This theorem is referenced by:  ordtbas2  21218  ordtrest2  21231  cnvordtrestixx  30290
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