Theorem cnvps 18538

Description: The converse of a poset is a poset. In the general case (^◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18539 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 6064	. . 3 ⊢ Rel ◡𝑅
2	1	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅)
3		cnvco 5835	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
4		pstr2 18531	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5		cnvss 5822	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
6	4, 5	syl 17	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
7	3, 6	eqsstrrid 3962	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
8		psrel 18529	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
9		dfrel2 6148	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
10	8, 9	sylib 218	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅)
11	10	ineq2d 4161	. . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅))
12		incom 4150	. . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅)
13	11, 12	eqtrdi 2788	. . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅))
14		psref2 18530	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
15		relcnvfld 6239	. . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
16	8, 15	syl 17	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
17	16	reseq2d 5939	. . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
18	13, 14, 17	3eqtrd 2776	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
19		cnvexg 7869	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V)
20		isps 18528	. . 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 1344	1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∪ cuni 4851   I cid 5519  ^◡ccnv 5624   ↾ cres 5627   ∘ ccom 5629  Rel wrel 5630  PosetRelcps 18524

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 2709  ax-sep 5232  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ps 18526

This theorem is referenced by:  cnvpsb  18539  cnvtsr  18548  ordtcnv  23179  xrge0iifhmeo  34099