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Theorem isps 17800
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 5644 . . 3 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 coeq1 5721 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑟))
3 coeq2 5722 . . . . 5 (𝑟 = 𝑅 → (𝑅𝑟) = (𝑅𝑅))
42, 3eqtrd 2853 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
5 id 22 . . . 4 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5sseq12d 3997 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
7 cnveq 5737 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
85, 7ineq12d 4187 . . . 4 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
9 unieq 4838 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝑅)
109unieqd 4840 . . . . 5 (𝑟 = 𝑅 𝑟 = 𝑅)
1110reseq2d 5846 . . . 4 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝑅))
128, 11eqeq12d 2834 . . 3 (𝑟 = 𝑅 → ((𝑟𝑟) = ( I ↾ 𝑟) ↔ (𝑅𝑅) = ( I ↾ 𝑅)))
131, 6, 123anbi123d 1427 . 2 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
14 df-ps 17798 . 2 PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟𝑟) ⊆ 𝑟 ∧ (𝑟𝑟) = ( I ↾ 𝑟))}
1513, 14elab2g 3665 1 (𝑅𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105  cin 3932  wss 3933   cuni 4830   I cid 5452  ccnv 5547  cres 5550  ccom 5552  Rel wrel 5553  PosetRelcps 17796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-in 3940  df-ss 3949  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-res 5560  df-ps 17798
This theorem is referenced by:  psrel  17801  psref2  17802  pstr2  17803  cnvps  17810  psss  17812  letsr  17825
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