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Mirrors > Home > ILE Home > Th. List > iswomni | GIF version |
Description: The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
Ref | Expression |
---|---|
iswomni | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | feq2 5315 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑓:𝑦⟶2o ↔ 𝑓:𝐴⟶2o)) | |
2 | raleq 2659 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) | |
3 | 2 | dcbid 828 | . . . 4 ⊢ (𝑦 = 𝐴 → (DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) |
4 | 1, 3 | imbi12d 233 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) ↔ (𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
5 | 4 | albidv 1811 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
6 | df-womni 7119 | . 2 ⊢ WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)} | |
7 | 5, 6 | elab2g 2868 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 DECID wdc 824 ∀wal 1340 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ⟶wf 5178 ‘cfv 5182 1oc1o 6368 2oc2o 6369 WOmnicwomni 7118 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-fn 5185 df-f 5186 df-womni 7119 |
This theorem is referenced by: iswomnimap 7121 omniwomnimkv 7122 |
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