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Theorem fxnn0nninf 10398
Description: A function from 0* into . (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7104 instead of infnninfOLD 7105. More generally, this theorem and most theorems in this section could use an extended 𝐺 defined by 𝐺 = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) ∪ ⟨ω, +∞⟩) and 𝐹 = (𝑛 ∈ suc ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅))) as in nnnninf2 7107.
Hypotheses
Ref Expression
fxnn0nninf.g 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
fxnn0nninf.f 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))
fxnn0nninf.i 𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
Assertion
Ref Expression
fxnn0nninf 𝐼:ℕ0*⟶ℕ
Distinct variable group:   𝑖,𝑛
Allowed substitution hints:   𝐹(𝑥,𝑖,𝑛)   𝐺(𝑥,𝑖,𝑛)   𝐼(𝑥,𝑖,𝑛)

Proof of Theorem fxnn0nninf
StepHypRef Expression
1 fxnn0nninf.g . . . . . 6 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
2 fxnn0nninf.f . . . . . 6 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))
31, 2fnn0nninf 10397 . . . . 5 (𝐹𝐺):ℕ0⟶ℕ
4 pnfex 7977 . . . . . . . 8 +∞ ∈ V
5 omex 4578 . . . . . . . . 9 ω ∈ V
6 1oex 6407 . . . . . . . . . 10 1o ∈ V
76snex 4172 . . . . . . . . 9 {1o} ∈ V
85, 7xpex 4727 . . . . . . . 8 (ω × {1o}) ∈ V
94, 8f1osn 5485 . . . . . . 7 {⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})}
10 f1of 5445 . . . . . . 7 ({⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})} → {⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})})
119, 10ax-mp 5 . . . . . 6 {⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})}
12 infnninfOLD 7105 . . . . . . 7 (ω × {1o}) ∈ ℕ
13 snssi 3725 . . . . . . 7 ((ω × {1o}) ∈ ℕ → {(ω × {1o})} ⊆ ℕ)
1412, 13ax-mp 5 . . . . . 6 {(ω × {1o})} ⊆ ℕ
15 fss 5361 . . . . . 6 (({⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})} ∧ {(ω × {1o})} ⊆ ℕ) → {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ)
1611, 14, 15mp2an 424 . . . . 5 {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ
173, 16pm3.2i 270 . . . 4 ((𝐹𝐺):ℕ0⟶ℕ ∧ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ)
18 disj 3464 . . . . 5 ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞})
19 nn0nepnf 9210 . . . . . . 7 (𝑥 ∈ ℕ0𝑥 ≠ +∞)
2019neneqd 2362 . . . . . 6 (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞)
21 elsni 3602 . . . . . 6 (𝑥 ∈ {+∞} → 𝑥 = +∞)
2220, 21nsyl 624 . . . . 5 (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞})
2318, 22mprgbir 2529 . . . 4 (ℕ0 ∩ {+∞}) = ∅
24 fun2 5373 . . . 4 ((((𝐹𝐺):ℕ0⟶ℕ ∧ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ) ∧ (ℕ0 ∩ {+∞}) = ∅) → ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ)
2517, 23, 24mp2an 424 . . 3 ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ
26 fxnn0nninf.i . . . 4 𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
2726feq1i 5342 . . 3 (𝐼:(ℕ0 ∪ {+∞})⟶ℕ ↔ ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ)
2825, 27mpbir 145 . 2 𝐼:(ℕ0 ∪ {+∞})⟶ℕ
29 df-xnn0 9203 . . 3 0* = (ℕ0 ∪ {+∞})
3029feq2i 5343 . 2 (𝐼:ℕ0*⟶ℕ𝐼:(ℕ0 ∪ {+∞})⟶ℕ)
3128, 30mpbir 145 1 𝐼:ℕ0*⟶ℕ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1349  wcel 2142  cun 3120  cin 3121  wss 3122  c0 3415  ifcif 3527  {csn 3584  cop 3587  cmpt 4051  ωcom 4575   × cxp 4610  ccnv 4611  ccom 4616  wf 5196  1-1-ontowf1o 5199  (class class class)co 5857  freccfrec 6373  1oc1o 6392  xnninf 7100  0cc0 7778  1c1 7779   + caddc 7781  +∞cpnf 7955  0cn0 9139  0*cxnn0 9202  cz 9216
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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-addcom 7878  ax-addass 7880  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-0id 7886  ax-rnegex 7887  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-ltadd 7894
This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-recs 6288  df-frec 6374  df-1o 6399  df-2o 6400  df-map 6632  df-nninf 7101  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-inn 8883  df-n0 9140  df-xnn0 9203  df-z 9217  df-uz 9492
This theorem is referenced by: (None)
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