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Mirrors > Home > ILE Home > Th. List > frecuzrdg0 | GIF version |
Description: Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10172 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 27-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
frecuzrdgrrn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
frecuzrdgrrn.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
frecuzrdgrrn.2 | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
frecuzrdgtcl.3 | ⊢ (𝜑 → 𝑇 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdg0 | ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frec2uz.2 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
3 | frecuzrdgrrn.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | frecuzrdgrrn.f | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | frecuzrdgrrn.2 | . . . 4 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
6 | frecuzrdgtcl.3 | . . . 4 ⊢ (𝜑 → 𝑇 = ran 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | frecuzrdgtcl 10185 | . . 3 ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆) |
8 | ffun 5275 | . . 3 ⊢ (𝑇:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑇) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → Fun 𝑇) |
10 | 5 | fveq1i 5422 | . . . . 5 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) |
11 | opexg 4150 | . . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → 〈𝐶, 𝐴〉 ∈ V) | |
12 | 1, 3, 11 | syl2anc 408 | . . . . . 6 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ V) |
13 | frec0g 6294 | . . . . . 6 ⊢ (〈𝐶, 𝐴〉 ∈ V → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) = 〈𝐶, 𝐴〉) | |
14 | 12, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) = 〈𝐶, 𝐴〉) |
15 | 10, 14 | syl5eq 2184 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = 〈𝐶, 𝐴〉) |
16 | 1, 2, 3, 4, 5 | frecuzrdgrcl 10183 | . . . . . 6 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
17 | ffn 5272 | . . . . . 6 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
18 | 16, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 Fn ω) |
19 | peano1 4508 | . . . . 5 ⊢ ∅ ∈ ω | |
20 | fnfvelrn 5552 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
21 | 18, 19, 20 | sylancl 409 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅) |
22 | 15, 21 | eqeltrrd 2217 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ ran 𝑅) |
23 | 22, 6 | eleqtrrd 2219 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑇) |
24 | funopfv 5461 | . 2 ⊢ (Fun 𝑇 → (〈𝐶, 𝐴〉 ∈ 𝑇 → (𝑇‘𝐶) = 𝐴)) | |
25 | 9, 23, 24 | sylc 62 | 1 ⊢ (𝜑 → (𝑇‘𝐶) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 〈cop 3530 ↦ cmpt 3989 ωcom 4504 × cxp 4537 ran crn 4540 Fun wfun 5117 Fn wfn 5118 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 freccfrec 6287 1c1 7621 + caddc 7623 ℤcz 9054 ℤ≥cuz 9326 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: (None) |
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