Theorem fxnn0nninf 10694

Description: A function from ℕ₀^* into ℕ_∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7317 instead of infnninfOLD 7318. 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(𝑖 ∈ 𝑛, 1_o, ∅))) as in nnnninf2 7320.

Hypotheses

Ref	Expression
fxnn0nninf.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
fxnn0nninf.f	⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
fxnn0nninf.i	⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (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

Step	Hyp	Ref	Expression
1		fxnn0nninf.g	. . . . . 6 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
2		fxnn0nninf.f	. . . . . 6 ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
3	1, 2	fnn0nninf 10693	. . . . 5 ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞
4		pnfex 8226	. . . . . . . 8 ⊢ +∞ ∈ V
5		omex 4689	. . . . . . . . 9 ⊢ ω ∈ V
6		1oex 6585	. . . . . . . . . 10 ⊢ 1o ∈ V
7	6	snex 4273	. . . . . . . . 9 ⊢ {1o} ∈ V
8	5, 7	xpex 4840	. . . . . . . 8 ⊢ (ω × {1o}) ∈ V
9	4, 8	f1osn 5621	. . . . . . 7 ⊢ {⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})}
10		f1of 5580	. . . . . . 7 ⊢ ({⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})} → {⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})})
11	9, 10	ax-mp 5	. . . . . 6 ⊢ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})}
12		infnninfOLD 7318	. . . . . . 7 ⊢ (ω × {1o}) ∈ ℕ∞
13		snssi 3815	. . . . . . 7 ⊢ ((ω × {1o}) ∈ ℕ∞ → {(ω × {1o})} ⊆ ℕ∞)
14	12, 13	ax-mp 5	. . . . . 6 ⊢ {(ω × {1o})} ⊆ ℕ∞
15		fss 5491	. . . . . 6 ⊢ (({⟨+∞, (ω × {1o})⟩}:{+∞}⟶{(ω × {1o})} ∧ {(ω × {1o})} ⊆ ℕ∞) → {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ∞)
16	11, 14, 15	mp2an 426	. . . . 5 ⊢ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ∞
17	3, 16	pm3.2i 272	. . . 4 ⊢ ((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ∞)
18		disj 3541	. . . . 5 ⊢ ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞})
19		nn0nepnf 9466	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ≠ +∞)
20	19	neneqd 2421	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞)
21		elsni 3685	. . . . . 6 ⊢ (𝑥 ∈ {+∞} → 𝑥 = +∞)
22	20, 21	nsyl 631	. . . . 5 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞})
23	18, 22	mprgbir 2588	. . . 4 ⊢ (ℕ0 ∩ {+∞}) = ∅
24		fun2 5506	. . . 4 ⊢ ((((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {⟨+∞, (ω × {1o})⟩}:{+∞}⟶ℕ∞) ∧ (ℕ0 ∩ {+∞}) = ∅) → ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ∞)
25	17, 23, 24	mp2an 426	. . 3 ⊢ ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ∞
26		fxnn0nninf.i	. . . 4 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
27	26	feq1i 5472	. . 3 ⊢ (𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ ↔ ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ∞)
28	25, 27	mpbir 146	. 2 ⊢ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞
29		df-xnn0 9459	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
30	29	feq2i 5473	. 2 ⊢ (𝐼:ℕ0*⟶ℕ∞ ↔ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞)
31	28, 30	mpbir 146	1 ⊢ 𝐼:ℕ0*⟶ℕ∞

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ∪ cun 3196   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492  ifcif 3603  {csn 3667  ⟨cop 3670   ↦ cmpt 4148  ωcom 4686   × cxp 4721  ^◡ccnv 4722   ∘ ccom 4727  ⟶wf 5320  –1-1-onto→wf1o 5323  (class class class)co 6013  freccfrec 6551  1_oc1o 6570  ℕ_∞xnninf 7312  0cc0 8025  1c1 8026   + caddc 8028  +∞cpnf 8204  ℕ₀cn0 9395  ℕ₀^*cxnn0 9458  ℤcz 9472

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-addass 8127  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-ltadd 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-recs 6466  df-frec 6552  df-1o 6577  df-2o 6578  df-map 6814  df-nninf 7313  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-inn 9137  df-n0 9396  df-xnn0 9459  df-z 9473  df-uz 9749

This theorem is referenced by:  nninfctlemfo  12604