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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 10348 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 10361 | . . 3 ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆) |
8 | ffun 5348 | . . 3 ⊢ (𝑇:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑇) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → Fun 𝑇) |
10 | 5 | fveq1i 5495 | . . . . 5 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) |
11 | opexg 4211 | . . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → 〈𝐶, 𝐴〉 ∈ V) | |
12 | 1, 3, 11 | syl2anc 409 | . . . . . 6 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ V) |
13 | frec0g 6374 | . . . . . 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 | eqtrid 2215 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = 〈𝐶, 𝐴〉) |
16 | 1, 2, 3, 4, 5 | frecuzrdgrcl 10359 | . . . . . 6 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
17 | ffn 5345 | . . . . . 6 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
18 | 16, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 Fn ω) |
19 | peano1 4576 | . . . . 5 ⊢ ∅ ∈ ω | |
20 | fnfvelrn 5626 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
21 | 18, 19, 20 | sylancl 411 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅) |
22 | 15, 21 | eqeltrrd 2248 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ ran 𝑅) |
23 | 22, 6 | eleqtrrd 2250 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑇) |
24 | funopfv 5534 | . 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 1348 ∈ wcel 2141 Vcvv 2730 ∅c0 3414 〈cop 3584 ↦ cmpt 4048 ωcom 4572 × cxp 4607 ran crn 4610 Fun wfun 5190 Fn wfn 5191 ⟶wf 5192 ‘cfv 5196 (class class class)co 5851 ∈ cmpo 5853 freccfrec 6367 1c1 7768 + caddc 7770 ℤcz 9205 ℤ≥cuz 9480 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 df-uz 9481 |
This theorem is referenced by: (None) |
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