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Mirrors > Home > ILE Home > Th. List > frecuzrdg0t | GIF version |
Description: Initial value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 28-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frecuzrdgrclt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
frecuzrdgrclt.t | ⊢ (𝜑 → 𝑆 ⊆ 𝑇) |
frecuzrdgrclt.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
frecuzrdgrclt.r | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
frecuzrdg0t.ran | ⊢ (𝜑 → 𝑃 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdg0t | ⊢ (𝜑 → (𝑃‘𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrclt.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frecuzrdgrclt.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | frecuzrdgrclt.t | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | |
4 | frecuzrdgrclt.f | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | frecuzrdgrclt.r | . . . 4 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
6 | frecuzrdg0t.ran | . . . 4 ⊢ (𝜑 → 𝑃 = ran 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | frecuzrdgtclt 9731 | . . 3 ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆) |
8 | ffun 5120 | . . 3 ⊢ (𝑃:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑃) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → Fun 𝑃) |
10 | 5 | fveq1i 5257 | . . . . 5 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) |
11 | opexg 4022 | . . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → 〈𝐶, 𝐴〉 ∈ V) | |
12 | 1, 2, 11 | syl2anc 403 | . . . . . 6 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ V) |
13 | frec0g 6097 | . . . . . 6 ⊢ (〈𝐶, 𝐴〉 ∈ V → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) = 〈𝐶, 𝐴〉) | |
14 | 12, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) = 〈𝐶, 𝐴〉) |
15 | 10, 14 | syl5eq 2129 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = 〈𝐶, 𝐴〉) |
16 | 1, 2, 3, 4, 5 | frecuzrdgrclt 9725 | . . . . . 6 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
17 | ffn 5117 | . . . . . 6 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
18 | 16, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 Fn ω) |
19 | peano1 4375 | . . . . 5 ⊢ ∅ ∈ ω | |
20 | fnfvelrn 5379 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
21 | 18, 19, 20 | sylancl 404 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅) |
22 | 15, 21 | eqeltrrd 2162 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ ran 𝑅) |
23 | 22, 6 | eleqtrrd 2164 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑃) |
24 | funopfv 5292 | . 2 ⊢ (Fun 𝑃 → (〈𝐶, 𝐴〉 ∈ 𝑃 → (𝑃‘𝐶) = 𝐴)) | |
25 | 9, 23, 24 | sylc 61 | 1 ⊢ (𝜑 → (𝑃‘𝐶) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1287 ∈ wcel 1436 Vcvv 2614 ⊆ wss 2986 ∅c0 3272 〈cop 3428 ωcom 4371 × cxp 4402 ran crn 4405 Fun wfun 4966 Fn wfn 4967 ⟶wf 4968 ‘cfv 4972 (class class class)co 5594 ↦ cmpt2 5596 freccfrec 6090 1c1 7272 + caddc 7274 ℤcz 8660 ℤ≥cuz 8928 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-iord 4160 df-on 4162 df-ilim 4163 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-frec 6091 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 df-uz 8929 |
This theorem is referenced by: iseq1t 9767 |
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