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Theorem isps 17249
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 5235 . . 3 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 coeq1 5312 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑟))
3 coeq2 5313 . . . . 5 (𝑟 = 𝑅 → (𝑅𝑟) = (𝑅𝑅))
42, 3eqtrd 2685 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5sseq12d 3667 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
7 cnveq 5328 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
85, 7ineq12d 3848 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
9 unieq 4476 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝑅)
109unieqd 4478 . . . . 5 (𝑟 = 𝑅 𝑟 = 𝑅)
1110reseq2d 5428 . . . 4 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝑅))
128, 11eqeq12d 2666 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) = ( I ↾ 𝑟) ↔ (𝑅𝑅) = ( I ↾ 𝑅)))
131, 6, 123anbi123d 1439 . 2 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
14 df-ps 17247 . 2 PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
1513, 14elab2g 3385 1 (𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054   = wceq 1523  wcel 2030  cin 3606  wss 3607   cuni 4468   I cid 5052  ccnv 5142  cres 5145  ccom 5147  Rel wrel 5148  PosetRelcps 17245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-res 5155  df-ps 17247
This theorem is referenced by:  psrel  17250  psref2  17251  pstr2  17252  cnvps  17259  psss  17261  letsr  17274
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