Theorem cnvtsr 18549

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 18548	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2		cnvps 18539	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
3	1, 2	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel)
4		eqid 2741	. . . . 5 ⊢ dom 𝑅 = dom 𝑅
5	4	istsr 18544	. . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
6	5	simprbi 499	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))
7	4	psrn 18536	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
8	1, 7	syl 17	. . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
9	8	sqxpeqd 5652	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10		psrel 18530	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
11	1, 10	syl 17	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅)
12		dfrel2 6143	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
13	11, 12	sylib 220	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅)
14	13	uneq2d 4100	. . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅))
15		uncom 4090	. . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅)
16	14, 15	eqtr2di 2793	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅))
17	6, 9, 16	3sstr3d 3970	. 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))
18		df-rn 5631	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
19	18	istsr 18544	. 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)))
20	3, 17, 19	sylanbrc 590	1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121   ∪ cun 3882   ⊆ wss 3884   × cxp 5618  ^◡ccnv 5619  dom cdm 5620  ran crn 5621  Rel wrel 5625  PosetRelcps 18525   TosetRel ctsr 18526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ps 18527  df-tsr 18528

This theorem is referenced by:  ordtbas2  23177  ordtrest2  23190  cnvordtrestixx  34107