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Theorem ismkv 7037
 Description: The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
Assertion
Ref Expression
ismkv (𝐴𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
Distinct variable group:   𝐴,𝑓,𝑥
Allowed substitution hints:   𝑉(𝑥,𝑓)

Proof of Theorem ismkv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 feq2 5265 . . . 4 (𝑦 = 𝐴 → (𝑓:𝑦⟶2o𝑓:𝐴⟶2o))
2 raleq 2630 . . . . . 6 (𝑦 = 𝐴 → (∀𝑥𝑦 (𝑓𝑥) = 1o ↔ ∀𝑥𝐴 (𝑓𝑥) = 1o))
32notbid 657 . . . . 5 (𝑦 = 𝐴 → (¬ ∀𝑥𝑦 (𝑓𝑥) = 1o ↔ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
4 rexeq 2631 . . . . 5 (𝑦 = 𝐴 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ↔ ∃𝑥𝐴 (𝑓𝑥) = ∅))
53, 4imbi12d 233 . . . 4 (𝑦 = 𝐴 → ((¬ ∀𝑥𝑦 (𝑓𝑥) = 1o → ∃𝑥𝑦 (𝑓𝑥) = ∅) ↔ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))
61, 5imbi12d 233 . . 3 (𝑦 = 𝐴 → ((𝑓:𝑦⟶2o → (¬ ∀𝑥𝑦 (𝑓𝑥) = 1o → ∃𝑥𝑦 (𝑓𝑥) = ∅)) ↔ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
76albidv 1797 . 2 (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥𝑦 (𝑓𝑥) = 1o → ∃𝑥𝑦 (𝑓𝑥) = ∅)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
8 df-markov 7036 . 2 Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥𝑦 (𝑓𝑥) = 1o → ∃𝑥𝑦 (𝑓𝑥) = ∅))}
97, 8elab2g 2836 1 (𝐴𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  ∀wal 1330   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  ∅c0 3369  ⟶wf 5128  ‘cfv 5132  1oc1o 6315  2oc2o 6316  Markovcmarkov 7035 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-fn 5135  df-f 5136  df-markov 7036 This theorem is referenced by:  ismkvmap  7038  omnimkv  7040  mkvprop  7042  omniwomnimkv  7051
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