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Mirrors > Home > ILE Home > Th. List > iswomnimap | GIF version |
Description: The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
Ref | Expression |
---|---|
iswomnimap | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iswomni 7120 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) | |
2 | 2onn 6480 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | elmapg 6618 | . . . . . 6 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o)) | |
4 | 2, 3 | mpan 421 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o)) |
5 | 4 | imbi1d 230 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ (𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
6 | 5 | albidv 1811 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
7 | 1, 6 | bitr4d 190 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
8 | df-ral 2447 | . 2 ⊢ (∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) | |
9 | 7, 8 | bitr4di 197 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)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 ωcom 4561 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 1oc1o 6368 2oc2o 6369 ↑𝑚 cmap 6605 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-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1o 6375 df-2o 6376 df-map 6607 df-womni 7119 |
This theorem is referenced by: enwomnilem 7124 iswomninnlem 13769 |
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