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Theorem isps 18525
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 5767 . . 3 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 coeq1 5848 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑟))
3 coeq2 5849 . . . . 5 (𝑟 = 𝑅 → (𝑅𝑟) = (𝑅𝑅))
42, 3eqtrd 2764 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5sseq12d 4008 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
7 cnveq 5864 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
85, 7ineq12d 4206 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
9 unieq 4911 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝑅)
109unieqd 4913 . . . . 5 (𝑟 = 𝑅 𝑟 = 𝑅)
1110reseq2d 5972 . . . 4 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝑅))
128, 11eqeq12d 2740 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) = ( I ↾ 𝑟) ↔ (𝑅𝑅) = ( I ↾ 𝑅)))
131, 6, 123anbi123d 1432 . 2 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
14 df-ps 18523 . 2 PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
1513, 14elab2g 3663 1 (𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  cin 3940  wss 3941   cuni 4900   I cid 5564  ccnv 5666  cres 5669  ccom 5671  Rel wrel 5672  PosetRelcps 18521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3948  df-ss 3958  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-res 5679  df-ps 18523
This theorem is referenced by:  psrel  18526  psref2  18527  pstr2  18528  cnvps  18535  psss  18537  letsr  18550
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