Theorem cnvps 18539

Description: The converse of a poset is a poset. In the general case (^◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18540 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 6063	. . 3 ⊢ Rel ◡𝑅
2	1	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅)
3		cnvco 5834	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
4		pstr2 18532	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5		cnvss 5817	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
6	4, 5	syl 17	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
7	3, 6	eqsstrrid 3956	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
8		psrel 18530	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
9		dfrel2 6144	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
10	8, 9	sylib 220	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅)
11	10	ineq2d 4152	. . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅))
12		incom 4141	. . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅)
13	11, 12	eqtrdi 2792	. . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅))
14		psref2 18531	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
15		relcnvfld 6235	. . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
16	8, 15	syl 17	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
17	16	reseq2d 5938	. . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
18	13, 14, 17	3eqtrd 2780	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
19		cnvexg 7868	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V)
20		isps 18529	. . 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 1350	1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  Vcvv 3433   ∩ cin 3884   ⊆ wss 3885  ∪ cuni 4841   I cid 5515  ^◡ccnv 5620   ↾ cres 5623   ∘ ccom 5625  Rel wrel 5626  PosetRelcps 18525

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 5221  ax-pow 5297  ax-pr 5365  ax-un 7682

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 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ps 18527

This theorem is referenced by:  cnvpsb  18540  cnvtsr  18549  ordtcnv  23188  xrge0iifhmeo  34132