Theorem isps 18614

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 5754	. . 3 ⊢ (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2		coeq1 5834	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑟))
3		coeq2 5835	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑅 ∘ 𝑟) = (𝑅 ∘ 𝑅))
4	2, 3	eqtrd 2800	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅))
5		id 23	. . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
6	4, 5	sseq12d 3972	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
7		cnveq 5850	. . . . 5 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
8	5, 7	ineq12d 4176	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∩ ◡𝑟) = (𝑅 ∩ ◡𝑅))
9		unieq 4879	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅)
10	9	unieqd 4881	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅)
11	10	reseq2d 5969	. . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟) = ( I ↾ ∪ ∪ 𝑅))
12	8, 11	eqeq12d 2781	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟) ↔ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))
13	1, 6, 12	3anbi123d 1460	. 2 ⊢ (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
14		df-ps 18612	. 2 ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
15	13, 14	elab2g 3642	1 ⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∩ cin 3906   ⊆ wss 3907  ∪ cuni 4868   I cid 5546  ^◡ccnv 5651   ↾ cres 5654   ∘ ccom 5656  Rel wrel 5657  PosetRelcps 18610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-in 3914  df-ss 3924  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-res 5664  df-ps 18612

This theorem is referenced by:  psrel  18615  psref2  18616  pstr2  18617  cnvps  18624  psss  18626  letsr  18639