![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > isomni | GIF version |
Description: The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.) |
Ref | Expression |
---|---|
isomni | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 5140 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2𝑜 ↔ 𝑓:𝐴⟶2𝑜)) | |
2 | rexeq 2563 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)) | |
3 | raleq 2562 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜 ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜)) | |
4 | 2, 3 | orbi12d 742 | . . . 4 ⊢ (𝑦 = 𝐴 → ((∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜) ↔ (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜))) |
5 | 1, 4 | imbi12d 232 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑓:𝑦⟶2𝑜 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜)) ↔ (𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜)))) |
6 | 5 | albidv 1752 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜)) ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜)))) |
7 | df-omni 6780 | . 2 ⊢ Omni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2𝑜 → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1𝑜))} | |
8 | 6, 7 | elab2g 2762 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2𝑜 → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1𝑜)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 664 ∀wal 1287 = wceq 1289 ∈ wcel 1438 ∀wral 2359 ∃wrex 2360 ∅c0 3286 ⟶wf 5006 ‘cfv 5010 1𝑜c1o 6166 2𝑜c2o 6167 Omnicomni 6778 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-fn 5013 df-f 5014 df-omni 6780 |
This theorem is referenced by: isomnimap 6783 finomni 6786 exmidomniim 6787 exmidomni 6788 |
Copyright terms: Public domain | W3C validator |