Theorem ismkv 7254

Description: The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)

Assertion

Ref	Expression
ismkv	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))

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.

Step	Hyp	Ref	Expression
1		feq2 5408	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o))
2		raleq 2701	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
3	2	notbid 668	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
4		rexeq 2702	. . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))
5	3, 4	imbi12d 234	. . . 4 ⊢ (𝑦 = 𝐴 → ((¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅) ↔ (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
6	1, 5	imbi12d 234	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅)) ↔ (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
7	6	albidv 1846	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
8		df-markov 7253	. 2 ⊢ Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅))}
9	7, 8	elab2g 2919	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1370   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484  ∅c0 3459  ⟶wf 5266  ‘cfv 5270  1_oc1o 6494  2_oc2o 6495  Markovcmarkov 7252

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-fn 5273  df-f 5274  df-markov 7253

This theorem is referenced by:  ismkvmap  7255  omnimkv  7257  mkvprop  7259  omniwomnimkv  7268