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Mirrors > Home > MPE Home > Th. List > om2uzrani | Structured version Visualization version GIF version |
Description: Range of 𝐺 (see om2uz0i 13065). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
om2uz.1 | ⊢ 𝐶 ∈ ℤ |
om2uz.2 | ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) |
Ref | Expression |
---|---|
om2uzrani | ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frfnom 7813 | . . . . . 6 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω | |
2 | om2uz.2 | . . . . . . 7 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
3 | 2 | fneq1i 6230 | . . . . . 6 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 223 | . . . . 5 ⊢ 𝐺 Fn ω |
5 | fvelrnb 6503 | . . . . 5 ⊢ (𝐺 Fn ω → (𝑦 ∈ ran 𝐺 ↔ ∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝑦 ∈ ran 𝐺 ↔ ∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦) |
7 | om2uz.1 | . . . . . . 7 ⊢ 𝐶 ∈ ℤ | |
8 | 7, 2 | om2uzuzi 13067 | . . . . . 6 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) |
9 | eleq1 2846 | . . . . . 6 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) ↔ 𝑦 ∈ (ℤ≥‘𝐶))) | |
10 | 8, 9 | syl5ibcom 237 | . . . . 5 ⊢ (𝑧 ∈ ω → ((𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶))) |
11 | 10 | rexlimiv 3208 | . . . 4 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶)) |
12 | 6, 11 | sylbi 209 | . . 3 ⊢ (𝑦 ∈ ran 𝐺 → 𝑦 ∈ (ℤ≥‘𝐶)) |
13 | eleq1 2846 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ ran 𝐺 ↔ 𝐶 ∈ ran 𝐺)) | |
14 | eleq1 2846 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐺 ↔ 𝑦 ∈ ran 𝐺)) | |
15 | eleq1 2846 | . . . 4 ⊢ (𝑧 = (𝑦 + 1) → (𝑧 ∈ ran 𝐺 ↔ (𝑦 + 1) ∈ ran 𝐺)) | |
16 | 7, 2 | om2uz0i 13065 | . . . . 5 ⊢ (𝐺‘∅) = 𝐶 |
17 | peano1 7363 | . . . . . 6 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6620 | . . . . . 6 ⊢ ((𝐺 Fn ω ∧ ∅ ∈ ω) → (𝐺‘∅) ∈ ran 𝐺) | |
19 | 4, 17, 18 | mp2an 682 | . . . . 5 ⊢ (𝐺‘∅) ∈ ran 𝐺 |
20 | 16, 19 | eqeltrri 2855 | . . . 4 ⊢ 𝐶 ∈ ran 𝐺 |
21 | 7, 2 | om2uzsuci 13066 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
22 | oveq1 6929 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) + 1) = (𝑦 + 1)) | |
23 | 21, 22 | sylan9eq 2833 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) = (𝑦 + 1)) |
24 | peano2 7364 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω) | |
25 | fnfvelrn 6620 | . . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ suc 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺) | |
26 | 4, 24, 25 | sylancr 581 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) ∈ ran 𝐺) |
27 | 26 | adantr 474 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) ∈ ran 𝐺) |
28 | 23, 27 | eqeltrrd 2859 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝑦 + 1) ∈ ran 𝐺) |
29 | 28 | rexlimiva 3209 | . . . . . 6 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺) |
30 | 6, 29 | sylbi 209 | . . . . 5 ⊢ (𝑦 ∈ ran 𝐺 → (𝑦 + 1) ∈ ran 𝐺) |
31 | 30 | a1i 11 | . . . 4 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → (𝑦 ∈ ran 𝐺 → (𝑦 + 1) ∈ ran 𝐺)) |
32 | 13, 14, 15, 14, 20, 31 | uzind4i 12056 | . . 3 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → 𝑦 ∈ ran 𝐺) |
33 | 12, 32 | impbii 201 | . 2 ⊢ (𝑦 ∈ ran 𝐺 ↔ 𝑦 ∈ (ℤ≥‘𝐶)) |
34 | 33 | eqriv 2774 | 1 ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∃wrex 3090 Vcvv 3397 ∅c0 4140 ↦ cmpt 4965 ran crn 5356 ↾ cres 5357 suc csuc 5978 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 ωcom 7343 reccrdg 7788 1c1 10273 + caddc 10275 ℤcz 11728 ℤ≥cuz 11992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 |
This theorem is referenced by: om2uzf1oi 13071 |
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