Theorem ismkv 7212

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 5387	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o))
2		raleq 2690	. . . . . 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 2691	. . . . 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 1835	. 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 7211	. 2 ⊢ Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅))}
9	7, 8	elab2g 2907	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 1362   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∅c0 3446  ⟶wf 5250  ‘cfv 5254  1_oc1o 6462  2_oc2o 6463  Markovcmarkov 7210

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-fn 5257  df-f 5258  df-markov 7211

This theorem is referenced by:  ismkvmap  7213  omnimkv  7215  mkvprop  7217  omniwomnimkv  7226