![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > frecuzrdg0 | GIF version |
Description: Initial value of a recursive definition generator on upper integers. See comment in frec2uz0d 10203 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 10216 | . . 3 ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆) |
8 | ffun 5283 | . . 3 ⊢ (𝑇:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑇) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → Fun 𝑇) |
10 | 5 | fveq1i 5430 | . . . . 5 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉)‘∅) |
11 | opexg 4158 | . . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → 〈𝐶, 𝐴〉 ∈ V) | |
12 | 1, 3, 11 | syl2anc 409 | . . . . . 6 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ V) |
13 | frec0g 6302 | . . . . . 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 2185 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) = 〈𝐶, 𝐴〉) |
16 | 1, 2, 3, 4, 5 | frecuzrdgrcl 10214 | . . . . . 6 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
17 | ffn 5280 | . . . . . 6 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
18 | 16, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 Fn ω) |
19 | peano1 4516 | . . . . 5 ⊢ ∅ ∈ ω | |
20 | fnfvelrn 5560 | . . . . 5 ⊢ ((𝑅 Fn ω ∧ ∅ ∈ ω) → (𝑅‘∅) ∈ ran 𝑅) | |
21 | 18, 19, 20 | sylancl 410 | . . . 4 ⊢ (𝜑 → (𝑅‘∅) ∈ ran 𝑅) |
22 | 15, 21 | eqeltrrd 2218 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ ran 𝑅) |
23 | 22, 6 | eleqtrrd 2220 | . 2 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ 𝑇) |
24 | funopfv 5469 | . 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 1332 ∈ wcel 1481 Vcvv 2689 ∅c0 3368 〈cop 3535 ↦ cmpt 3997 ωcom 4512 × cxp 4545 ran crn 4548 Fun wfun 5125 Fn wfn 5126 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 freccfrec 6295 1c1 7645 + caddc 7647 ℤcz 9078 ℤ≥cuz 9350 |
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-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-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-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 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-z 9079 df-uz 9351 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |