Theorem cnvps 18612

Description: The converse of a poset is a poset. In the general case (^◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18613 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
cnvps	⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)

Proof of Theorem cnvps

Step	Hyp	Ref	Expression
1		relcnv 6095	. . 3 ⊢ Rel ◡𝑅
2	1	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅)
3		cnvco 5863	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
4		pstr2 18605	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5		cnvss 5846	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
6	4, 5	syl 17	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
7	3, 6	eqsstrrid 3977	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
8		psrel 18603	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
9		dfrel2 6177	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
10	8, 9	sylib 220	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅)
11	10	ineq2d 4174	. . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅))
12		incom 4163	. . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅)
13	11, 12	eqtrdi 2815	. . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅))
14		psref2 18604	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
15		relcnvfld 6269	. . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
16	8, 15	syl 17	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
17	16	reseq2d 5967	. . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
18	13, 14, 17	3eqtrd 2803	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
19		cnvexg 7907	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V)
20		isps 18602	. . 3 ⊢ (◡𝑅 ∈ V → (◡𝑅 ∈ PosetRel ↔ (Rel ◡𝑅 ∧ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 ∧ (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))))
21	19, 20	syl 17	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∈ PosetRel ↔ (Rel ◡𝑅 ∧ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 ∧ (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))))
22	2, 7, 18, 21	mpbir3and 1357	1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  Vcvv 3456   ∩ cin 3905   ⊆ wss 3906  ∪ cuni 4867   I cid 5543  ^◡ccnv 5648   ↾ cres 5651   ∘ ccom 5653  Rel wrel 5654  PosetRelcps 18598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ps 18600

This theorem is referenced by:  cnvpsb  18613  cnvtsr  18622  ordtcnv  23263  xrge0iifhmeo  34235