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Theorem isps 18613
Description: The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
Assertion
Ref Expression
isps (𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))

Proof of Theorem isps
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 releq 5786 . . 3 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 coeq1 5868 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑟))
3 coeq2 5869 . . . . 5 (𝑟 = 𝑅 → (𝑅𝑟) = (𝑅𝑅))
42, 3eqtrd 2777 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5sseq12d 4017 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
7 cnveq 5884 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
85, 7ineq12d 4221 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
9 unieq 4918 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝑅)
109unieqd 4920 . . . . 5 (𝑟 = 𝑅 𝑟 = 𝑅)
1110reseq2d 5997 . . . 4 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝑅))
128, 11eqeq12d 2753 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) = ( I ↾ 𝑟) ↔ (𝑅𝑅) = ( I ↾ 𝑅)))
131, 6, 123anbi123d 1438 . 2 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
14 df-ps 18611 . 2 PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
1513, 14elab2g 3680 1 (𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951   cuni 4907   I cid 5577  ccnv 5684  cres 5687  ccom 5689  Rel wrel 5690  PosetRelcps 18609
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-res 5697  df-ps 18611
This theorem is referenced by:  psrel  18614  psref2  18615  pstr2  18616  cnvps  18623  psss  18625  letsr  18638
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