Theorem cnvtsr 17045

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 17044	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2		cnvps 17035	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
3	1, 2	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel)
4		eqid 2610	. . . . 5 ⊢ dom 𝑅 = dom 𝑅
5	4	istsr 17040	. . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
6	5	simprbi 479	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))
7	4	psrn 17032	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
8	1, 7	syl 17	. . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
9	8	sqxpeqd 5065	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10		psrel 17026	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
11	1, 10	syl 17	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅)
12		dfrel2 5502	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
13	11, 12	sylib 207	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅)
14	13	uneq2d 3729	. . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅))
15		uncom 3719	. . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅)
16	14, 15	syl6req 2661	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅))
17	6, 9, 16	3sstr3d 3610	. 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))
18		df-rn 5049	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
19	18	istsr 17040	. 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)))
20	3, 17, 19	sylanbrc 695	1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  Rel wrel 5043  PosetRelcps 17021   TosetRel ctsr 17022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ps 17023  df-tsr 17024

This theorem is referenced by:  ordtbas2  20805  ordtrest2  20818  cnvordtrestixx  29287