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Mirrors > Home > ILE Home > Th. List > frecuzrdgtclt | GIF version |
Description: The recursive definition generator on upper integers is a function. (Contributed by Jim Kingdon, 22-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〉) |
frecuzrdgtclt.3 | ⊢ (𝜑 → 𝑃 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdgtclt | ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrclt.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frecuzrdgrclt.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | frecuzrdgrclt.t | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | |
4 | frecuzrdgrclt.f | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | frecuzrdgrclt.r | . . . . 5 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
6 | 1, 2, 3, 4, 5 | frecuzrdgfun 10186 | . . . 4 ⊢ (𝜑 → Fun ran 𝑅) |
7 | frecuzrdgtclt.3 | . . . . 5 ⊢ (𝜑 → 𝑃 = ran 𝑅) | |
8 | 7 | funeqd 5140 | . . . 4 ⊢ (𝜑 → (Fun 𝑃 ↔ Fun ran 𝑅)) |
9 | 6, 8 | mpbird 166 | . . 3 ⊢ (𝜑 → Fun 𝑃) |
10 | 7 | dmeqd 4736 | . . . 4 ⊢ (𝜑 → dom 𝑃 = dom ran 𝑅) |
11 | 1, 2, 3, 4, 5 | frecuzrdgdom 10184 | . . . 4 ⊢ (𝜑 → dom ran 𝑅 = (ℤ≥‘𝐶)) |
12 | 10, 11 | eqtrd 2170 | . . 3 ⊢ (𝜑 → dom 𝑃 = (ℤ≥‘𝐶)) |
13 | df-fn 5121 | . . 3 ⊢ (𝑃 Fn (ℤ≥‘𝐶) ↔ (Fun 𝑃 ∧ dom 𝑃 = (ℤ≥‘𝐶))) | |
14 | 9, 12, 13 | sylanbrc 413 | . 2 ⊢ (𝜑 → 𝑃 Fn (ℤ≥‘𝐶)) |
15 | 1, 2, 3, 4, 5 | frecuzrdgrclt 10181 | . . . 4 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
16 | frn 5276 | . . . 4 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆)) | |
17 | 15, 16 | syl 14 | . . 3 ⊢ (𝜑 → ran 𝑅 ⊆ ((ℤ≥‘𝐶) × 𝑆)) |
18 | 7, 17 | eqsstrd 3128 | . 2 ⊢ (𝜑 → 𝑃 ⊆ ((ℤ≥‘𝐶) × 𝑆)) |
19 | dff2 5557 | . 2 ⊢ (𝑃:(ℤ≥‘𝐶)⟶𝑆 ↔ (𝑃 Fn (ℤ≥‘𝐶) ∧ 𝑃 ⊆ ((ℤ≥‘𝐶) × 𝑆))) | |
20 | 14, 18, 19 | sylanbrc 413 | 1 ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ⊆ wss 3066 〈cop 3525 ωcom 4499 × cxp 4532 dom cdm 4534 ran crn 4535 Fun wfun 5112 Fn wfn 5113 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 ∈ cmpo 5769 freccfrec 6280 1c1 7614 + caddc 7616 ℤcz 9047 ℤ≥cuz 9319 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 |
This theorem is referenced by: frecuzrdg0t 10188 frecuzrdgsuctlem 10189 seqf 10227 seqf2 10230 |
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