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Theorem isps 17403
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 5358 . . 3 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 coeq1 5435 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑟))
3 coeq2 5436 . . . . 5 (𝑟 = 𝑅 → (𝑅𝑟) = (𝑅𝑅))
42, 3eqtrd 2794 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5sseq12d 3775 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
7 cnveq 5451 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
85, 7ineq12d 3958 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
9 unieq 4596 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝑅)
109unieqd 4598 . . . . 5 (𝑟 = 𝑅 𝑟 = 𝑅)
1110reseq2d 5551 . . . 4 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝑅))
128, 11eqeq12d 2775 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) = ( I ↾ 𝑟) ↔ (𝑅𝑅) = ( I ↾ 𝑅)))
131, 6, 123anbi123d 1548 . 2 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
14 df-ps 17401 . 2 PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
1513, 14elab2g 3493 1 (𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  wcel 2139  cin 3714  wss 3715   cuni 4588   I cid 5173  ccnv 5265  cres 5268  ccom 5270  Rel wrel 5271  PosetRelcps 17399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-res 5278  df-ps 17401
This theorem is referenced by:  psrel  17404  psref2  17405  pstr2  17406  cnvps  17413  psss  17415  letsr  17428
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