Theorem iswomni 7364

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 5466	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o))
2		raleq 2730	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
3	2	dcbid 845	. . . 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 1872	. 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 7363	. 2 ⊢ WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)}
7	5, 6	elab2g 2953	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 841  ∀wal 1395   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ⟶wf 5322  ‘cfv 5326  1_oc1o 6575  2_oc2o 6576  WOmnicwomni 7362

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-fn 5329  df-f 5330  df-womni 7363

This theorem is referenced by:  iswomnimap  7365  omniwomnimkv  7366