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Mirrors > Home > ILE Home > Th. List > frecuzrdgcl | GIF version |
Description: Closure law for the recursive definition generator on upper integers. See comment in frec2uz0d 9401 for the description of 𝐺 as the mapping from ω to (ℤ≥‘𝐶). (Contributed by Jim Kingdon, 31-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
uzrdg.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
uzrdg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
uzrdg.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
uzrdg.2 | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
frecuzrdgfn.3 | ⊢ (𝜑 → 𝑇 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdgcl | ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘𝐵) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | 1 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ) |
3 | frec2uz.2 | . . . . 5 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
4 | uzrdg.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 4 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑆 ∈ 𝑉) |
6 | uzrdg.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
7 | 6 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆) |
8 | uzrdg.f | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
9 | 8 | adantlr 460 | . . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
10 | uzrdg.2 | . . . . 5 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
11 | simpr 108 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐵 ∈ (ℤ≥‘𝐶)) | |
12 | 2, 3, 5, 7, 9, 10, 11 | frecuzrdglem 9413 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
13 | frecuzrdgfn.3 | . . . . 5 ⊢ (𝜑 → 𝑇 = ran 𝑅) | |
14 | 13 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑇 = ran 𝑅) |
15 | 12, 14 | eleqtrrd 2158 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ 𝑇) |
16 | 1, 3, 4, 6, 8, 10, 13 | frecuzrdgfn 9414 | . . . . 5 ⊢ (𝜑 → 𝑇 Fn (ℤ≥‘𝐶)) |
17 | fnfun 5016 | . . . . 5 ⊢ (𝑇 Fn (ℤ≥‘𝐶) → Fun 𝑇) | |
18 | funopfv 5234 | . . . . 5 ⊢ (Fun 𝑇 → (〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ 𝑇 → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))) | |
19 | 16, 17, 18 | 3syl 17 | . . . 4 ⊢ (𝜑 → (〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ 𝑇 → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
20 | 19 | adantr 270 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ 𝑇 → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
21 | 15, 20 | mpd 13 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))) |
22 | 1, 3 | frec2uzf1od 9408 | . . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
23 | f1ocnvdm 5441 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω) | |
24 | 22, 23 | sylan 277 | . . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω) |
25 | 1, 3, 4, 6, 8, 10 | frecuzrdgrrn 9410 | . . . 4 ⊢ ((𝜑 ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆)) |
26 | 24, 25 | syldan 276 | . . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆)) |
27 | xp2nd 5813 | . . 3 ⊢ ((𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆) | |
28 | 26, 27 | syl 14 | . 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆) |
29 | 21, 28 | eqeltrd 2155 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘𝐵) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 〈cop 3401 ↦ cmpt 3839 ωcom 4331 × cxp 4361 ◡ccnv 4362 ran crn 4364 Fun wfun 4916 Fn wfn 4917 –1-1-onto→wf1o 4921 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 2nd c2nd 5786 freccfrec 6000 1c1 6982 + caddc 6984 ℤcz 8351 ℤ≥cuz 8619 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 |
This theorem is referenced by: iseqcl 9443 |
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