Theorem fxnn0nninf 10548

Description: A function from ℕ₀^* into ℕ_∞. (Contributed by Jim Kingdon, 16-Jul-2022.) TODO: use infnninf 7199 instead of infnninfOLD 7200. 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 7202.

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 10547	. . . . 5 ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞
4		pnfex 8097	. . . . . . . 8 ⊢ +∞ ∈ V
5		omex 4630	. . . . . . . . 9 ⊢ ω ∈ V
6		1oex 6491	. . . . . . . . . 10 ⊢ 1o ∈ V
7	6	snex 4219	. . . . . . . . 9 ⊢ {1o} ∈ V
8	5, 7	xpex 4779	. . . . . . . 8 ⊢ (ω × {1o}) ∈ V
9	4, 8	f1osn 5547	. . . . . . 7 ⊢ {⟨+∞, (ω × {1o})⟩}:{+∞}–1-1-onto→{(ω × {1o})}
10		f1of 5507	. . . . . . 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 7200	. . . . . . 7 ⊢ (ω × {1o}) ∈ ℕ∞
13		snssi 3767	. . . . . . 7 ⊢ ((ω × {1o}) ∈ ℕ∞ → {(ω × {1o})} ⊆ ℕ∞)
14	12, 13	ax-mp 5	. . . . . 6 ⊢ {(ω × {1o})} ⊆ ℕ∞
15		fss 5422	. . . . . 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 3500	. . . . 5 ⊢ ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞})
19		nn0nepnf 9337	. . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ≠ +∞)
20	19	neneqd 2388	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞)
21		elsni 3641	. . . . . 6 ⊢ (𝑥 ∈ {+∞} → 𝑥 = +∞)
22	20, 21	nsyl 629	. . . . 5 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞})
23	18, 22	mprgbir 2555	. . . 4 ⊢ (ℕ0 ∩ {+∞}) = ∅
24		fun2 5434	. . . 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 5403	. . 3 ⊢ (𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ ↔ ((𝐹 ∘ ◡𝐺) ∪ {⟨+∞, (ω × {1o})⟩}):(ℕ0 ∪ {+∞})⟶ℕ∞)
28	25, 27	mpbir 146	. 2 ⊢ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞
29		df-xnn0 9330	. . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞})
30	29	feq2i 5404	. 2 ⊢ (𝐼:ℕ0*⟶ℕ∞ ↔ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞)
31	28, 30	mpbir 146	1 ⊢ 𝐼:ℕ0*⟶ℕ∞

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157  ∅c0 3451  ifcif 3562  {csn 3623  ⟨cop 3626   ↦ cmpt 4095  ωcom 4627   × cxp 4662  ^◡ccnv 4663   ∘ ccom 4668  ⟶wf 5255  –1-1-onto→wf1o 5258  (class class class)co 5925  freccfrec 6457  1_oc1o 6476  ℕ_∞xnninf 7194  0cc0 7896  1c1 7897   + caddc 7899  +∞cpnf 8075  ℕ₀cn0 9266  ℕ₀^*cxnn0 9329  ℤcz 9343

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-map 6718  df-nninf 7195  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-xnn0 9330  df-z 9344  df-uz 9619

This theorem is referenced by:  nninfctlemfo  12232