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Mirrors > Home > ILE Home > Th. List > fxnn0nninf | GIF version |
Description: A function from ℕ0* into ℕ∞. (Contributed by Jim Kingdon, 16-Jul-2022.) |
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})〉}) |
Ref | Expression |
---|---|
fxnn0nninf | ⊢ 𝐼:ℕ0*⟶ℕ∞ |
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 10241 | . . . . 5 ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ |
4 | pnfex 7843 | . . . . . . . 8 ⊢ +∞ ∈ V | |
5 | omex 4515 | . . . . . . . . 9 ⊢ ω ∈ V | |
6 | 1oex 6329 | . . . . . . . . . 10 ⊢ 1o ∈ V | |
7 | 6 | snex 4117 | . . . . . . . . 9 ⊢ {1o} ∈ V |
8 | 5, 7 | xpex 4662 | . . . . . . . 8 ⊢ (ω × {1o}) ∈ V |
9 | 4, 8 | f1osn 5415 | . . . . . . 7 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}–1-1-onto→{(ω × {1o})} |
10 | f1of 5375 | . . . . . . 7 ⊢ ({〈+∞, (ω × {1o})〉}:{+∞}–1-1-onto→{(ω × {1o})} → {〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})}) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})} |
12 | infnninf 7030 | . . . . . . 7 ⊢ (ω × {1o}) ∈ ℕ∞ | |
13 | snssi 3672 | . . . . . . 7 ⊢ ((ω × {1o}) ∈ ℕ∞ → {(ω × {1o})} ⊆ ℕ∞) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ {(ω × {1o})} ⊆ ℕ∞ |
15 | fss 5292 | . . . . . 6 ⊢ (({〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})} ∧ {(ω × {1o})} ⊆ ℕ∞) → {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) | |
16 | 11, 14, 15 | mp2an 423 | . . . . 5 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞ |
17 | 3, 16 | pm3.2i 270 | . . . 4 ⊢ ((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) |
18 | disj 3416 | . . . . 5 ⊢ ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞}) | |
19 | nn0nepnf 9072 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ≠ +∞) | |
20 | 19 | neneqd 2330 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞) |
21 | elsni 3550 | . . . . . 6 ⊢ (𝑥 ∈ {+∞} → 𝑥 = +∞) | |
22 | 20, 21 | nsyl 618 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞}) |
23 | 18, 22 | mprgbir 2493 | . . . 4 ⊢ (ℕ0 ∩ {+∞}) = ∅ |
24 | fun2 5304 | . . . 4 ⊢ ((((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) ∧ (ℕ0 ∩ {+∞}) = ∅) → ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞) | |
25 | 17, 23, 24 | mp2an 423 | . . 3 ⊢ ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞ |
26 | fxnn0nninf.i | . . . 4 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) | |
27 | 26 | feq1i 5273 | . . 3 ⊢ (𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ ↔ ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞) |
28 | 25, 27 | mpbir 145 | . 2 ⊢ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ |
29 | df-xnn0 9065 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
30 | 29 | feq2i 5274 | . 2 ⊢ (𝐼:ℕ0*⟶ℕ∞ ↔ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞) |
31 | 28, 30 | mpbir 145 | 1 ⊢ 𝐼:ℕ0*⟶ℕ∞ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ∪ cun 3074 ∩ cin 3075 ⊆ wss 3076 ∅c0 3368 ifcif 3479 {csn 3532 〈cop 3535 ↦ cmpt 3997 ωcom 4512 × cxp 4545 ◡ccnv 4546 ∘ ccom 4551 ⟶wf 5127 –1-1-onto→wf1o 5130 (class class class)co 5782 freccfrec 6295 1oc1o 6314 ℕ∞xnninf 7013 0cc0 7644 1c1 7645 + caddc 7647 +∞cpnf 7821 ℕ0cn0 9001 ℕ0*cxnn0 9064 ℤcz 9078 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-recs 6210 df-frec 6296 df-1o 6321 df-2o 6322 df-map 6552 df-nninf 7015 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-xnn0 9065 df-z 9079 df-uz 9351 |
This theorem is referenced by: (None) |
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