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Theorem isomni 6782
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 5140 . . . 4 (𝑦 = 𝐴 → (𝑓:𝑦⟶2𝑜𝑓:𝐴⟶2𝑜))
2 rexeq 2563 . . . . 5 (𝑦 = 𝐴 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ↔ ∃𝑥𝐴 (𝑓𝑥) = ∅))
3 raleq 2562 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝑦 (𝑓𝑥) = 1𝑜 ↔ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜))
42, 3orbi12d 742 . . . 4 (𝑦 = 𝐴 → ((∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜) ↔ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜)))
51, 4imbi12d 232 . . 3 (𝑦 = 𝐴 → ((𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)) ↔ (𝑓:𝐴⟶2𝑜 → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜))))
65albidv 1752 . 2 (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜)) ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜))))
7 df-omni 6780 . 2 Omni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1𝑜))}
86, 7elab2g 2762 1 (𝐴𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1𝑜))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 664  wal 1287   = wceq 1289  wcel 1438  wral 2359  wrex 2360  c0 3286  wf 5006  cfv 5010  1𝑜c1o 6166  2𝑜c2o 6167  Omnicomni 6778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-fn 5013  df-f 5014  df-omni 6780
This theorem is referenced by:  isomnimap  6783  finomni  6786  exmidomniim  6787  exmidomni  6788
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