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

Step	Hyp	Ref	Expression
1		tsrps 18632	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2		cnvps 18623	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
3	1, 2	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel)
4		eqid 2737	. . . . 5 ⊢ dom 𝑅 = dom 𝑅
5	4	istsr 18628	. . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
6	5	simprbi 496	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))
7	4	psrn 18620	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
8	1, 7	syl 17	. . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
9	8	sqxpeqd 5717	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10		psrel 18614	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
11	1, 10	syl 17	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅)
12		dfrel2 6209	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
13	11, 12	sylib 218	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅)
14	13	uneq2d 4168	. . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅))
15		uncom 4158	. . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅)
16	14, 15	eqtr2di 2794	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅))
17	6, 9, 16	3sstr3d 4038	. 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))
18		df-rn 5696	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
19	18	istsr 18628	. 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)))
20	3, 17, 19	sylanbrc 583	1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )

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