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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) |
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〉) |
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 | 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 | 1, 2 | frec2uzf1od 9702 | . . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
7 | frecuzrdglem.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶)) | |
8 | f1ocnvdm 5500 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω) | |
9 | 6, 7, 8 | syl2anc 403 | . . . 4 ⊢ (𝜑 → (◡𝐺‘𝐵) ∈ ω) |
10 | 1, 2, 3, 4, 5, 9 | frec2uzrdg 9705 | . . 3 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
11 | f1ocnvfv2 5497 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) | |
12 | 6, 7, 11 | syl2anc 403 | . . . 4 ⊢ (𝜑 → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) |
13 | 12 | opeq1d 3602 | . . 3 ⊢ (𝜑 → 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
14 | 10, 13 | eqtrd 2115 | . 2 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
15 | 1, 2, 3, 4, 5 | frecuzrdgrcl 9706 | . . . 4 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
16 | ffn 5114 | . . . 4 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
17 | 15, 16 | syl 14 | . . 3 ⊢ (𝜑 → 𝑅 Fn ω) |
18 | fnfvelrn 5376 | . . 3 ⊢ ((𝑅 Fn ω ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) | |
19 | 17, 9, 18 | syl2anc 403 | . 2 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) |
20 | 14, 19 | eqeltrrd 2160 | 1 ⊢ (𝜑 → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 〈cop 3425 ↦ cmpt 3865 ωcom 4368 × cxp 4399 ◡ccnv 4400 ran crn 4402 Fn wfn 4964 ⟶wf 4965 –1-1-onto→wf1o 4968 ‘cfv 4969 (class class class)co 5591 ↦ cmpt2 5593 2nd c2nd 5845 freccfrec 6087 1c1 7254 + caddc 7256 ℤcz 8646 ℤ≥cuz 8914 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 |
This theorem is referenced by: frecuzrdgtcl 9708 frecuzrdgsuc 9710 |
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