Theorem cnvps 18713

Description: The converse of a poset is a poset. In the general case (^◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18714 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 6094	. . 3 ⊢ Rel ◡𝑅
2	1	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅)
3		cnvco 5863	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
4		pstr2 18706	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5		cnvss 5846	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
6	4, 5	syl 18	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
7	3, 6	eqsstrrid 3969	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
8		psrel 18704	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
9		dfrel2 6176	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
10	8, 9	sylib 221	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅)
11	10	ineq2d 4165	. . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅))
12		incom 4154	. . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅)
13	11, 12	eqtrdi 2811	. . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅))
14		psref2 18705	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
15		relcnvfld 6272	. . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
16	8, 15	syl 18	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
17	16	reseq2d 5966	. . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
18	13, 14, 17	3eqtrd 2799	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
19		cnvexg 7919	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V)
20		isps 18703	. . 3 ⊢ (◡𝑅 ∈ V → (◡𝑅 ∈ PosetRel ↔ (Rel ◡𝑅 ∧ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 ∧ (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))))
21	19, 20	syl 18	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∈ PosetRel ↔ (Rel ◡𝑅 ∧ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 ∧ (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))))
22	2, 7, 18, 21	mpbir3and 1361	1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4866   I cid 5541  ^◡ccnv 5646   ↾ cres 5649   ∘ ccom 5651  Rel wrel 5652  PosetRelcps 18699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ps 18701

This theorem is used by:  cnvpsb  18714  cnvtsr  18723  ordtcnv  23480  xrge0iifhmeo  34501