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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 5428 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑓‘suc 𝑖) = (𝐴‘suc 𝑖)) | |
2 | fveq1 5428 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑖) = (𝐴‘𝑖)) | |
3 | 1, 2 | sseq12d 3133 | . . . . 5 ⊢ (𝑓 = 𝐴 → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
4 | 3 | ralbidv 2438 | . . . 4 ⊢ (𝑓 = 𝐴 → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
5 | df-nninf 7015 | . . . 4 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
6 | 4, 5 | elrab2 2847 | . . 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 6572 | . 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 1332 ∈ wcel 1481 ∀wral 2417 ⊆ wss 3076 suc csuc 4295 ωcom 4512 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 2oc2o 6315 ↑𝑚 cmap 6550 ℕ∞xnninf 7013 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-map 6552 df-nninf 7015 |
This theorem is referenced by: nnsf 13374 peano4nninf 13375 nninfalllemn 13377 nninfall 13379 nninfsellemeqinf 13387 |
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