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Mirrors > Home > ILE Home > Th. List > 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 5514 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑓‘suc 𝑖) = (𝐴‘suc 𝑖)) | |
2 | fveq1 5514 | . . . . . 6 ⊢ (𝑓 = 𝐴 → (𝑓‘𝑖) = (𝐴‘𝑖)) | |
3 | 1, 2 | sseq12d 3186 | . . . . 5 ⊢ (𝑓 = 𝐴 → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
4 | 3 | ralbidv 2477 | . . . 4 ⊢ (𝑓 = 𝐴 → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
5 | df-nninf 7118 | . . . 4 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} | |
6 | 4, 5 | elrab2 2896 | . . 3 ⊢ (𝐴 ∈ ℕ∞ ↔ (𝐴 ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω (𝐴‘suc 𝑖) ⊆ (𝐴‘𝑖))) |
7 | 6 | simplbi 274 | . 2 ⊢ (𝐴 ∈ ℕ∞ → 𝐴 ∈ (2o ↑𝑚 ω)) |
8 | elmapi 6669 | . 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 1353 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 suc csuc 4365 ωcom 4589 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 2oc2o 6410 ↑𝑚 cmap 6647 ℕ∞xnninf 7117 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-map 6649 df-nninf 7118 |
This theorem is referenced by: nnnninfeq 7125 nnnninfeq2 7126 nninfisol 7130 nninfdcinf 7168 nninfwlpor 7171 nnsf 14636 peano4nninf 14637 nninfall 14640 nninfsellemeqinf 14647 |
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