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Mirrors > Home > ILE Home > Th. List > ismkv | GIF version |
Description: The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
Ref | Expression |
---|---|
ismkv | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 5256 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o)) | |
2 | raleq 2626 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) | |
3 | 2 | notbid 656 | . . . . 5 ⊢ (𝑦 = 𝐴 → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) |
4 | rexeq 2627 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)) | |
5 | 3, 4 | imbi12d 233 | . . . 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 233 | . . 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 1796 | . 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 7026 | . 2 ⊢ Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅))} | |
9 | 7, 8 | elab2g 2831 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ∅c0 3363 ⟶wf 5119 ‘cfv 5123 1oc1o 6306 2oc2o 6307 Markovcmarkov 7025 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-fn 5126 df-f 5127 df-markov 7026 |
This theorem is referenced by: ismkvmap 7028 omnimkv 7030 mkvprop 7032 |
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