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Mirrors > Home > MPE Home > Th. List > om2uzrani | Structured version Visualization version GIF version |
Description: Range of 𝐺 (see om2uz0i 12953). (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 7682 | . . . . . 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 6125 | . . . . . 6 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 221 | . . . . 5 ⊢ 𝐺 Fn ω |
5 | fvelrnb 6385 | . . . . 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 12955 | . . . . . 6 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) |
9 | eleq1 2837 | . . . . . 6 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) ↔ 𝑦 ∈ (ℤ≥‘𝐶))) | |
10 | 8, 9 | syl5ibcom 235 | . . . . 5 ⊢ (𝑧 ∈ ω → ((𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶))) |
11 | 10 | rexlimiv 3174 | . . . 4 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶)) |
12 | 6, 11 | sylbi 207 | . . 3 ⊢ (𝑦 ∈ ran 𝐺 → 𝑦 ∈ (ℤ≥‘𝐶)) |
13 | eleq1 2837 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ ran 𝐺 ↔ 𝐶 ∈ ran 𝐺)) | |
14 | eleq1 2837 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐺 ↔ 𝑦 ∈ ran 𝐺)) | |
15 | eleq1 2837 | . . . 4 ⊢ (𝑧 = (𝑦 + 1) → (𝑧 ∈ ran 𝐺 ↔ (𝑦 + 1) ∈ ran 𝐺)) | |
16 | 7, 2 | om2uz0i 12953 | . . . . 5 ⊢ (𝐺‘∅) = 𝐶 |
17 | peano1 7231 | . . . . . 6 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6499 | . . . . . 6 ⊢ ((𝐺 Fn ω ∧ ∅ ∈ ω) → (𝐺‘∅) ∈ ran 𝐺) | |
19 | 4, 17, 18 | mp2an 664 | . . . . 5 ⊢ (𝐺‘∅) ∈ ran 𝐺 |
20 | 16, 19 | eqeltrri 2846 | . . . 4 ⊢ 𝐶 ∈ ran 𝐺 |
21 | 7, 2 | om2uzsuci 12954 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
22 | oveq1 6799 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) + 1) = (𝑦 + 1)) | |
23 | 21, 22 | sylan9eq 2824 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) = (𝑦 + 1)) |
24 | peano2 7232 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω) | |
25 | fnfvelrn 6499 | . . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ suc 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺) | |
26 | 4, 24, 25 | sylancr 567 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) ∈ ran 𝐺) |
27 | 26 | adantr 466 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) ∈ ran 𝐺) |
28 | 23, 27 | eqeltrrd 2850 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝑦 + 1) ∈ ran 𝐺) |
29 | 28 | rexlimiva 3175 | . . . . . 6 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺) |
30 | 6, 29 | sylbi 207 | . . . . 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 11951 | . . 3 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → 𝑦 ∈ ran 𝐺) |
33 | 12, 32 | impbii 199 | . 2 ⊢ (𝑦 ∈ ran 𝐺 ↔ 𝑦 ∈ (ℤ≥‘𝐶)) |
34 | 33 | eqriv 2767 | 1 ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ∃wrex 3061 Vcvv 3349 ∅c0 4061 ↦ cmpt 4861 ran crn 5250 ↾ cres 5251 suc csuc 5868 Fn wfn 6026 ‘cfv 6031 (class class class)co 6792 ωcom 7211 reccrdg 7657 1c1 10138 + caddc 10140 ℤcz 11578 ℤ≥cuz 11887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-n0 11494 df-z 11579 df-uz 11888 |
This theorem is referenced by: om2uzf1oi 12959 |
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