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Theorem fxnn0nninf 10341
Description: A function from 0* into . (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7068 instead of infnninfOLD 7069. 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 7071.
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 10340 . . . . 5 (𝐹𝐺):ℕ0⟶ℕ
4 pnfex 7932 . . . . . . . 8 +∞ ∈ V
5 omex 4553 . . . . . . . . 9 ω ∈ V
6 1oex 6372 . . . . . . . . . 10 1o ∈ V
76snex 4147 . . . . . . . . 9 {1o} ∈ V
85, 7xpex 4702 . . . . . . . 8 (ω × {1o}) ∈ V
94, 8f1osn 5455 . . . . . . 7 {⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})}
10 f1of 5415 . . . . . . 7 ({⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})} → {⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})})
119, 10ax-mp 5 . . . . . 6 {⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})}
12 infnninfOLD 7069 . . . . . . 7 (ω × {1o}) ∈ ℕ
13 snssi 3701 . . . . . . 7 ((ω × {1o}) ∈ ℕ → {(ω × {1o})} ⊆ ℕ)
1412, 13ax-mp 5 . . . . . 6 {(ω × {1o})} ⊆ ℕ
15 fss 5332 . . . . . 6 (({⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})} ∧ {(ω × {1o})} ⊆ ℕ) → {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ)
1611, 14, 15mp2an 423 . . . . 5 {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ
173, 16pm3.2i 270 . . . 4 ((𝐹𝐺):ℕ0⟶ℕ ∧ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ)
18 disj 3442 . . . . 5 ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞})
19 nn0nepnf 9162 . . . . . . 7 (𝑥 ∈ ℕ0𝑥 ≠ +∞)
2019neneqd 2348 . . . . . 6 (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞)
21 elsni 3578 . . . . . 6 (𝑥 ∈ {+∞} → 𝑥 = +∞)
2220, 21nsyl 618 . . . . 5 (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞})
2318, 22mprgbir 2515 . . . 4 (ℕ0 ∩ {+∞}) = ∅
24 fun2 5344 . . . 4 ((((𝐹𝐺):ℕ0⟶ℕ ∧ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ) ∧ (ℕ0 ∩ {+∞}) = ∅) → ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ)
2517, 23, 24mp2an 423 . . 3 ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ
26 fxnn0nninf.i . . . 4 𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
2726feq1i 5313 . . 3 (𝐼:(ℕ0 ∪ {+∞})⟶ℕ ↔ ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ)
2825, 27mpbir 145 . 2 𝐼:(ℕ0 ∪ {+∞})⟶ℕ
29 df-xnn0 9155 . . 3 0* = (ℕ0 ∪ {+∞})
3029feq2i 5314 . 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 1335  wcel 2128  cun 3100  cin 3101  wss 3102  c0 3394  ifcif 3505  {csn 3560  cop 3563  cmpt 4026  ωcom 4550   × cxp 4585  ccnv 4586  ccom 4591  wf 5167  1-1-ontowf1o 5170  (class class class)co 5825  freccfrec 6338  1oc1o 6357  xnninf 7064  0cc0 7733  1c1 7734   + caddc 7736  +∞cpnf 7910  0cn0 9091  0*cxnn0 9154  cz 9168
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-addcom 7833  ax-addass 7835  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-0id 7841  ax-rnegex 7842  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-ltadd 7849
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-recs 6253  df-frec 6339  df-1o 6364  df-2o 6365  df-map 6596  df-nninf 7065  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-inn 8835  df-n0 9092  df-xnn0 9155  df-z 9169  df-uz 9441
This theorem is referenced by: (None)
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