Theorem cnvtsr 17222

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 17221	. . 3 ⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
2		cnvps 17212	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
3	1, 2	syl 17	. 2 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ PosetRel)
4		eqid 2622	. . . . 5 ⊢ dom 𝑅 = dom 𝑅
5	4	istsr 17217	. . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅)))
6	5	simprbi 480	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) ⊆ (𝑅 ∪ ◡𝑅))
7	4	psrn 17209	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
8	1, 7	syl 17	. . . 4 ⊢ (𝑅 ∈ TosetRel → dom 𝑅 = ran 𝑅)
9	8	sqxpeqd 5141	. . 3 ⊢ (𝑅 ∈ TosetRel → (dom 𝑅 × dom 𝑅) = (ran 𝑅 × ran 𝑅))
10		psrel 17203	. . . . . . 7 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
11	1, 10	syl 17	. . . . . 6 ⊢ (𝑅 ∈ TosetRel → Rel 𝑅)
12		dfrel2 5583	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
13	11, 12	sylib 208	. . . . 5 ⊢ (𝑅 ∈ TosetRel → ◡◡𝑅 = 𝑅)
14	13	uneq2d 3767	. . . 4 ⊢ (𝑅 ∈ TosetRel → (◡𝑅 ∪ ◡◡𝑅) = (◡𝑅 ∪ 𝑅))
15		uncom 3757	. . . 4 ⊢ (◡𝑅 ∪ 𝑅) = (𝑅 ∪ ◡𝑅)
16	14, 15	syl6req 2673	. . 3 ⊢ (𝑅 ∈ TosetRel → (𝑅 ∪ ◡𝑅) = (◡𝑅 ∪ ◡◡𝑅))
17	6, 9, 16	3sstr3d 3647	. 2 ⊢ (𝑅 ∈ TosetRel → (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅))
18		df-rn 5125	. . 3 ⊢ ran 𝑅 = dom ◡𝑅
19	18	istsr 17217	. 2 ⊢ (◡𝑅 ∈ TosetRel ↔ (◡𝑅 ∈ PosetRel ∧ (ran 𝑅 × ran 𝑅) ⊆ (◡𝑅 ∪ ◡◡𝑅)))
20	3, 17, 19	sylanbrc 698	1 ⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115  Rel wrel 5119  PosetRelcps 17198   TosetRel ctsr 17199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ps 17200  df-tsr 17201

This theorem is referenced by:  ordtbas2  20995  ordtrest2  21008  cnvordtrestixx  29959