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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 10419 | . . . . 5 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶)) |
7 | frecuzrdglem.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘𝐶)) | |
8 | f1ocnvdm 5795 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω) | |
9 | 6, 7, 8 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (◡𝐺‘𝐵) ∈ ω) |
10 | 1, 2, 3, 4, 5, 9 | frec2uzrdg 10422 | . . 3 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
11 | f1ocnvfv2 5792 | . . . . 5 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) | |
12 | 6, 7, 11 | syl2anc 411 | . . . 4 ⊢ (𝜑 → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) |
13 | 12 | opeq1d 3796 | . . 3 ⊢ (𝜑 → 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
14 | 10, 13 | eqtrd 2220 | . 2 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) = 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
15 | 1, 2, 3, 4, 5 | frecuzrdgrcl 10423 | . . . 4 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆)) |
16 | ffn 5377 | . . . 4 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → 𝑅 Fn ω) | |
17 | 15, 16 | syl 14 | . . 3 ⊢ (𝜑 → 𝑅 Fn ω) |
18 | fnfvelrn 5661 | . . 3 ⊢ ((𝑅 Fn ω ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) | |
19 | 17, 9, 18 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅) |
20 | 14, 19 | eqeltrrd 2265 | 1 ⊢ (𝜑 → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 〈cop 3607 ↦ cmpt 4076 ωcom 4601 × cxp 4636 ◡ccnv 4637 ran crn 4639 Fn wfn 5223 ⟶wf 5224 –1-1-onto→wf1o 5227 ‘cfv 5228 (class class class)co 5888 ∈ cmpo 5890 2nd c2nd 6153 freccfrec 6404 1c1 7825 + caddc 7827 ℤcz 9266 ℤ≥cuz 9541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 |
This theorem is referenced by: frecuzrdgtcl 10425 frecuzrdgsuc 10427 |
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