Theorem isps 18583

Description: The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)

Assertion

Ref	Expression
isps	⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))

Proof of Theorem isps

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		releq 5747	. . 3 ⊢ (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2		coeq1 5827	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑟))
3		coeq2 5828	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑅 ∘ 𝑟) = (𝑅 ∘ 𝑅))
4	2, 3	eqtrd 2796	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅))
5		id 22	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6	4, 5	sseq12d 3969	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
7		cnveq 5843	. . . . 5 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
8	5, 7	ineq12d 4173	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∩ ◡𝑟) = (𝑅 ∩ ◡𝑅))
9		unieq 4875	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅)
10	9	unieqd 4877	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅)
11	10	reseq2d 5963	. . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟) = ( I ↾ ∪ ∪ 𝑅))
12	8, 11	eqeq12d 2777	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟) ↔ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
13	1, 6, 12	3anbi123d 1456	. 2 ⊢ (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
14		df-ps 18581	. 2 ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
15	13, 14	elab2g 3639	1 ⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∩ cin 3903   ⊆ wss 3904  ∪ cuni 4864   I cid 5539  ^◡ccnv 5644   ↾ cres 5647   ∘ ccom 5649  Rel wrel 5650  PosetRelcps 18579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-in 3911  df-ss 3921  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-res 5657  df-ps 18581

This theorem is referenced by:  psrel  18584  psref2  18585  pstr2  18586  cnvps  18593  psss  18595  letsr  18608