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Theorem isomni 7058
 Description: The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
Assertion
Ref Expression
isomni (𝐴𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
Distinct variable group:   𝐴,𝑓,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑓)

Proof of Theorem isomni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 feq2 5296 . . . 4 (𝑦 = 𝐴 → (𝑓:𝑦⟶2o𝑓:𝐴⟶2o))
2 rexeq 2650 . . . . 5 (𝑦 = 𝐴 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ↔ ∃𝑥𝐴 (𝑓𝑥) = ∅))
3 raleq 2649 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑦 (𝑓𝑥) = 1o ↔ ∀𝑥𝐴 (𝑓𝑥) = 1o))
42, 3orbi12d 783 . . . 4 (𝑦 = 𝐴 → ((∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o) ↔ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)))
51, 4imbi12d 233 . . 3 (𝑦 = 𝐴 → ((𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)) ↔ (𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
65albidv 1801 . 2 (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
7 df-omni 7057 . 2 Omni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o))}
86, 7elab2g 2855 1 (𝐴𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 698  ∀wal 1330   = wceq 1332   ∈ wcel 2125  ∀wral 2432  ∃wrex 2433  ∅c0 3390  ⟶wf 5159  ‘cfv 5163  1oc1o 6346  2oc2o 6347  Omnicomni 7056 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-fn 5166  df-f 5167  df-omni 7057 This theorem is referenced by:  isomnimap  7059  finomni  7062  exmidomniim  7063  exmidomni  7064  omnimkv  7078  omniwomnimkv  7089
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