Theorem cnvtsr 18513

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 18512	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2		cnvps 18503	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
3	1, 2	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel)
4		eqid 2735	. . . . 5 ⊢ dom 𝑅 = dom 𝑅
5	4	istsr 18508	. . . 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 18500	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
8	1, 7	syl 17	. . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
9	8	sqxpeqd 5655	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10		psrel 18494	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
11	1, 10	syl 17	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅)
12		dfrel2 6146	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
13	11, 12	sylib 218	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅)
14	13	uneq2d 4119	. . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅))
15		uncom 4109	. . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅)
16	14, 15	eqtr2di 2787	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅))
17	6, 9, 16	3sstr3d 3987	. 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))
18		df-rn 5634	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
19	18	istsr 18508	. 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)))
20	3, 17, 19	sylanbrc 584	1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∪ cun 3898   ⊆ wss 3900   × cxp 5621  ^◡ccnv 5622  dom cdm 5623  ran crn 5624  Rel wrel 5628  PosetRelcps 18489   TosetRel ctsr 18490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ps 18491  df-tsr 18492

This theorem is referenced by:  ordtbas2  23137  ordtrest2  23150  cnvordtrestixx  34049