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Mirrors > Home > ILE Home > Th. List > frecuzrdglem | GIF version |
Description: A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 26-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〉) |
frecuzrdglem.b | ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶)) |
Ref | Expression |
---|---|
frecuzrdglem | ⊢ (𝜑 → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frec2uz.2 | . . . 4 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
3 | uzrdg.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | uzrdg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
5 | uzrdg.f | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
6 | uzrdg.2 | . . . 4 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
7 | 1, 2 | frec2uzf1od 9192 | . . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
8 | frecuzrdglem.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶)) | |
9 | f1ocnvdm 5421 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω) | |
10 | 7, 8, 9 | syl2anc 391 | . . . 4 ⊢ (𝜑 → (◡𝐺‘𝐵) ∈ ω) |
11 | 1, 2, 3, 4, 5, 6, 10 | frec2uzrdg 9195 | . . 3 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
12 | f1ocnvfv2 5418 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) | |
13 | 7, 8, 12 | syl2anc 391 | . . . 4 ⊢ (𝜑 → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) |
14 | 13 | opeq1d 3555 | . . 3 ⊢ (𝜑 → 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
15 | 11, 14 | eqtrd 2072 | . 2 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
16 | 1, 2, 3, 4, 5, 6 | frecuzrdgrom 9196 | . . 3 ⊢ (𝜑 → 𝑅 Fn ω) |
17 | fnfvelrn 5299 | . . 3 ⊢ ((𝑅 Fn ω ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) | |
18 | 16, 10, 17 | syl2anc 391 | . 2 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) |
19 | 15, 18 | eqeltrrd 2115 | 1 ⊢ (𝜑 → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 〈cop 3378 ↦ cmpt 3818 ωcom 4313 ◡ccnv 4344 ran crn 4346 Fn wfn 4897 –1-1-onto→wf1o 4901 ‘cfv 4902 (class class class)co 5512 ↦ cmpt2 5514 2nd c2nd 5766 freccfrec 5977 1c1 6890 + caddc 6892 ℤcz 8245 ℤ≥cuz 8473 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 |
This theorem is referenced by: frecuzrdgfn 9198 frecuzrdgcl 9199 frecuzrdgsuc 9201 |
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