Theorem cnvps 17825

Description: The converse of a poset is a poset. In the general case (^◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 17826 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 5970	. . 3 ⊢ Rel ◡𝑅
2	1	a1i 11	. 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅)
3		cnvco 5759	. . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅)
4		pstr2 17818	. . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5		cnvss 5746	. . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
6	4, 5	syl 17	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅)
7	3, 6	eqsstrrid 4019	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅)
8		psrel 17816	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅)
9		dfrel2 6049	. . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
10	8, 9	sylib 220	. . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅)
11	10	ineq2d 4192	. . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅))
12		incom 4181	. . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅)
13	11, 12	syl6eq 2875	. . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅))
14		psref2 17817	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
15		relcnvfld 6134	. . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
16	8, 15	syl 17	. . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅)
17	16	reseq2d 5856	. . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
18	13, 14, 17	3eqtrd 2863	. 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅))
19		cnvexg 7632	. . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V)
20		isps 17815	. . 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 1338	1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  ∪ cuni 4841   I cid 5462  ^◡ccnv 5557   ↾ cres 5560   ∘ ccom 5562  Rel wrel 5563  PosetRelcps 17811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ps 17813

This theorem is referenced by:  cnvpsb  17826  cnvtsr  17835  ordtcnv  21812  xrge0iifhmeo  31183