Theorem isomni 6782

Description: The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)

Assertion

Ref	Expression
isomni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 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.

Step	Hyp	Ref	Expression
1		feq2 5140	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2𝑜 ↔ 𝑓:𝐴⟶2𝑜))
2		rexeq 2563	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
3		raleq 2562	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜))
4	2, 3	orbi12d 742	. . . 4 ⊢ (𝑦 = 𝐴 → ((∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜) ↔ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜)))
5	1, 4	imbi12d 232	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑓:𝑦⟶2𝑜 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜)) ↔ (𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜))))
6	5	albidv 1752	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜)) ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜))))
7		df-omni 6780	. 2 ⊢ Omni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜))}
8	6, 7	elab2g 2762	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜))))

Syntax hints:   → wi 4   ↔ wb 103   ∨ wo 664  ∀wal 1287   = wceq 1289   ∈ wcel 1438  ∀wral 2359  ∃wrex 2360  ∅c0 3286  ⟶wf 5006  ‘cfv 5010  1_𝑜c1o 6166  2_𝑜c2o 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