Theorem isomni 6958

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 5214	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o))
2		rexeq 2601	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
3		raleq 2600	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
4	2, 3	orbi12d 765	. . . 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 233	. . 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 1778	. 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 6956	. 2 ⊢ Omni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o))}
8	6, 7	elab2g 2800	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 680  ∀wal 1312   = wceq 1314   ∈ wcel 1463  ∀wral 2390  ∃wrex 2391  ∅c0 3329  ⟶wf 5077  ‘cfv 5081  1_oc1o 6260  2_oc2o 6261  Omnicomni 6954

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-fn 5084  df-f 5085  df-omni 6956

This theorem is referenced by:  isomnimap  6959  finomni  6962  exmidomniim  6963  exmidomni  6964  omnimkv  6980