Theorem iswomni 7455

Description: The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)

Assertion

Ref	Expression
iswomni	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))

Distinct variable group:   𝐴,𝑓,𝑥

Allowed substitution hints:   𝑉(𝑥,𝑓)

Proof of Theorem iswomni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq2 5491	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o))
2		raleq 2740	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
3	2	dcbid 846	. . . 4 ⊢ (𝑦 = 𝐴 → (DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
4	1, 3	imbi12d 234	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) ↔ (𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
5	4	albidv 1873	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
6		df-womni 7454	. 2 ⊢ WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)}
7	5, 6	elab2g 2963	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 842  ∀wal 1396   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ⟶wf 5347  ‘cfv 5351  1_oc1o 6639  2_oc2o 6640  WOmnicwomni 7453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2814  df-fn 5354  df-f 5355  df-womni 7454

This theorem is referenced by:  iswomnimap  7456  omniwomnimkv  7457