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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninff | GIF version |
Description: An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
Ref | Expression |
---|---|
nninff | ⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5388 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑓‘suc 𝑖) = (𝐴‘suc 𝑖)) | |
2 | fveq1 5388 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑖) = (𝐴‘𝑖)) | |
3 | 1, 2 | sseq12d 3098 | . . . . 5 ⊢ (𝑓 = 𝐴 → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
4 | 3 | ralbidv 2414 | . . . 4 ⊢ (𝑓 = 𝐴 → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
5 | df-nninf 6975 | . . . 4 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
6 | 4, 5 | elrab2 2816 | . . 3 ⊢ (𝐴 ∈ ℕ∞ ↔ (𝐴 ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
7 | 6 | simplbi 272 | . 2 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 ∈ (2o ↑𝑚 ω)) |
8 | elmapi 6532 | . 2 ⊢ (𝐴 ∈ (2o ↑𝑚 ω) → 𝐴:ω⟶2o) | |
9 | 7, 8 | syl 14 | 1 ⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ∀wral 2393 ⊆ wss 3041 suc csuc 4257 ωcom 4474 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 2oc2o 6275 ↑𝑚 cmap 6510 ℕ∞xnninf 6973 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-map 6512 df-nninf 6975 |
This theorem is referenced by: nnsf 13126 peano4nninf 13127 nninfalllemn 13129 nninfall 13131 nninfsellemeqinf 13139 |
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