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Theorem infnninf 7153
Description: The point at infinity in is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (ω × {1o}), as fconstmpt 4691 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
Assertion
Ref Expression
infnninf (𝑖 ∈ ω ↦ 1o) ∈ ℕ

Proof of Theorem infnninf
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1lt2o 6468 . . . . . 6 1o ∈ 2o
21a1i 9 . . . . 5 ((⊤ ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
32fmpttd 5692 . . . 4 (⊤ → (𝑖 ∈ ω ↦ 1o):ω⟶2o)
43mptru 1373 . . 3 (𝑖 ∈ ω ↦ 1o):ω⟶2o
5 2on 6451 . . . 4 2o ∈ On
6 omex 4610 . . . 4 ω ∈ V
7 elmapg 6688 . . . 4 ((2o ∈ On ∧ ω ∈ V) → ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o))
85, 6, 7mp2an 426 . . 3 ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ↔ (𝑖 ∈ ω ↦ 1o):ω⟶2o)
94, 8mpbir 146 . 2 (𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω)
10 peano2 4612 . . . . . 6 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
11 eqidd 2190 . . . . . . 7 (𝑖 = suc 𝑗 → 1o = 1o)
12 eqid 2189 . . . . . . 7 (𝑖 ∈ ω ↦ 1o) = (𝑖 ∈ ω ↦ 1o)
13 1oex 6450 . . . . . . 7 1o ∈ V
1411, 12, 13fvmpt 5614 . . . . . 6 (suc 𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
1510, 14syl 14 . . . . 5 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = 1o)
16 eqidd 2190 . . . . . 6 (𝑖 = 𝑗 → 1o = 1o)
1716, 12, 13fvmpt 5614 . . . . 5 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘𝑗) = 1o)
1815, 17eqtr4d 2225 . . . 4 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
19 eqimss 3224 . . . 4 (((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗) → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2018, 19syl 14 . . 3 (𝑗 ∈ ω → ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2120rgen 2543 . 2 𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)
22 fveq1 5533 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓‘suc 𝑗) = ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗))
23 fveq1 5533 . . . . 5 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (𝑓𝑗) = ((𝑖 ∈ ω ↦ 1o)‘𝑗))
2422, 23sseq12d 3201 . . . 4 (𝑓 = (𝑖 ∈ ω ↦ 1o) → ((𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
2524ralbidv 2490 . . 3 (𝑓 = (𝑖 ∈ ω ↦ 1o) → (∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗) ↔ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
26 df-nninf 7150 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑗 ∈ ω (𝑓‘suc 𝑗) ⊆ (𝑓𝑗)}
2725, 26elrab2 2911 . 2 ((𝑖 ∈ ω ↦ 1o) ∈ ℕ ↔ ((𝑖 ∈ ω ↦ 1o) ∈ (2o𝑚 ω) ∧ ∀𝑗 ∈ ω ((𝑖 ∈ ω ↦ 1o)‘suc 𝑗) ⊆ ((𝑖 ∈ ω ↦ 1o)‘𝑗)))
289, 21, 27mpbir2an 944 1 (𝑖 ∈ ω ↦ 1o) ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wtru 1365  wcel 2160  wral 2468  Vcvv 2752  wss 3144  cmpt 4079  Oncon0 4381  suc csuc 4383  ωcom 4607  wf 5231  cfv 5235  (class class class)co 5897  1oc1o 6435  2oc2o 6436  𝑚 cmap 6675  xnninf 7149
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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1o 6442  df-2o 6443  df-map 6677  df-nninf 7150
This theorem is referenced by:  nnnninf2  7156  nninfwlpoimlemdc  7206  nninffeq  15248
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