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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 10178 | . . . . 5 ⊢ (𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ |
4 | pnfex 7787 | . . . . . . . 8 ⊢ +∞ ∈ V | |
5 | omex 4477 | . . . . . . . . 9 ⊢ ω ∈ V | |
6 | 1oex 6289 | . . . . . . . . . 10 ⊢ 1o ∈ V | |
7 | 6 | snex 4079 | . . . . . . . . 9 ⊢ {1o} ∈ V |
8 | 5, 7 | xpex 4624 | . . . . . . . 8 ⊢ (ω × {1o}) ∈ V |
9 | 4, 8 | f1osn 5375 | . . . . . . 7 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}–1-1-onto→{(ω × {1o})} |
10 | f1of 5335 | . . . . . . 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 6990 | . . . . . . 7 ⊢ (ω × {1o}) ∈ ℕ∞ | |
13 | snssi 3634 | . . . . . . 7 ⊢ ((ω × {1o}) ∈ ℕ∞ → {(ω × {1o})} ⊆ ℕ∞) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ {(ω × {1o})} ⊆ ℕ∞ |
15 | fss 5254 | . . . . . 6 ⊢ (({〈+∞, (ω × {1o})〉}:{+∞}⟶{(ω × {1o})} ∧ {(ω × {1o})} ⊆ ℕ∞) → {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) | |
16 | 11, 14, 15 | mp2an 422 | . . . . 5 ⊢ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞ |
17 | 3, 16 | pm3.2i 270 | . . . 4 ⊢ ((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) |
18 | disj 3381 | . . . . 5 ⊢ ((ℕ0 ∩ {+∞}) = ∅ ↔ ∀𝑥 ∈ ℕ0 ¬ 𝑥 ∈ {+∞}) | |
19 | nn0nepnf 9016 | . . . . . . 7 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ≠ +∞) | |
20 | 19 | neneqd 2306 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 = +∞) |
21 | elsni 3515 | . . . . . 6 ⊢ (𝑥 ∈ {+∞} → 𝑥 = +∞) | |
22 | 20, 21 | nsyl 602 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → ¬ 𝑥 ∈ {+∞}) |
23 | 18, 22 | mprgbir 2467 | . . . 4 ⊢ (ℕ0 ∩ {+∞}) = ∅ |
24 | fun2 5266 | . . . 4 ⊢ ((((𝐹 ∘ ◡𝐺):ℕ0⟶ℕ∞ ∧ {〈+∞, (ω × {1o})〉}:{+∞}⟶ℕ∞) ∧ (ℕ0 ∩ {+∞}) = ∅) → ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞) | |
25 | 17, 23, 24 | mp2an 422 | . . 3 ⊢ ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞ |
26 | fxnn0nninf.i | . . . 4 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) | |
27 | 26 | feq1i 5235 | . . 3 ⊢ (𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ ↔ ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}):(ℕ0 ∪ {+∞})⟶ℕ∞) |
28 | 25, 27 | mpbir 145 | . 2 ⊢ 𝐼:(ℕ0 ∪ {+∞})⟶ℕ∞ |
29 | df-xnn0 9009 | . . 3 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
30 | 29 | feq2i 5236 | . 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 1316 ∈ wcel 1465 ∪ cun 3039 ∩ cin 3040 ⊆ wss 3041 ∅c0 3333 ifcif 3444 {csn 3497 〈cop 3500 ↦ cmpt 3959 ωcom 4474 × cxp 4507 ◡ccnv 4508 ∘ ccom 4513 ⟶wf 5089 –1-1-onto→wf1o 5092 (class class class)co 5742 freccfrec 6255 1oc1o 6274 ℕ∞xnninf 6973 0cc0 7588 1c1 7589 + caddc 7591 +∞cpnf 7765 ℕ0cn0 8945 ℕ0*cxnn0 9008 ℤcz 9022 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-recs 6170 df-frec 6256 df-1o 6281 df-2o 6282 df-map 6512 df-nninf 6975 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-xnn0 9009 df-z 9023 df-uz 9295 |
This theorem is referenced by: (None) |
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