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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 7224 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) | |
2 | 2onn 6574 | . . . . . 6 ⊢ 2o ∈ ω | |
3 | elmapg 6715 | . . . . . 6 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o)) | |
4 | 2, 3 | mpan 424 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o)) |
5 | 4 | imbi1d 231 | . . . 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 1835 | . . 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 191 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))) |
8 | df-ral 2477 | . 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 198 | 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 105 DECID wdc 835 ∀wal 1362 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ωcom 4622 ⟶wf 5250 ‘cfv 5254 (class class class)co 5918 1oc1o 6462 2oc2o 6463 ↑𝑚 cmap 6702 WOmnicwomni 7222 |
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-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-id 4324 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1o 6469 df-2o 6470 df-map 6704 df-womni 7223 |
This theorem is referenced by: enwomnilem 7228 nninfdcinf 7230 nninfwlporlem 7232 nninfwlpoim 7237 iswomninnlem 15539 |
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