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Mirrors > Home > MPE Home > Th. List > om2uzrani | Structured version Visualization version GIF version |
Description: Range of 𝐺 (see om2uz0i 13316). (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 8070 | . . . . . 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 6450 | . . . . . 6 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) |
4 | 1, 3 | mpbir 233 | . . . . 5 ⊢ 𝐺 Fn ω |
5 | fvelrnb 6726 | . . . . 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 13318 | . . . . . 6 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) |
9 | eleq1 2900 | . . . . . 6 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) ↔ 𝑦 ∈ (ℤ≥‘𝐶))) | |
10 | 8, 9 | syl5ibcom 247 | . . . . 5 ⊢ (𝑧 ∈ ω → ((𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶))) |
11 | 10 | rexlimiv 3280 | . . . 4 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶)) |
12 | 6, 11 | sylbi 219 | . . 3 ⊢ (𝑦 ∈ ran 𝐺 → 𝑦 ∈ (ℤ≥‘𝐶)) |
13 | eleq1 2900 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ ran 𝐺 ↔ 𝐶 ∈ ran 𝐺)) | |
14 | eleq1 2900 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐺 ↔ 𝑦 ∈ ran 𝐺)) | |
15 | eleq1 2900 | . . . 4 ⊢ (𝑧 = (𝑦 + 1) → (𝑧 ∈ ran 𝐺 ↔ (𝑦 + 1) ∈ ran 𝐺)) | |
16 | 7, 2 | om2uz0i 13316 | . . . . 5 ⊢ (𝐺‘∅) = 𝐶 |
17 | peano1 7601 | . . . . . 6 ⊢ ∅ ∈ ω | |
18 | fnfvelrn 6848 | . . . . . 6 ⊢ ((𝐺 Fn ω ∧ ∅ ∈ ω) → (𝐺‘∅) ∈ ran 𝐺) | |
19 | 4, 17, 18 | mp2an 690 | . . . . 5 ⊢ (𝐺‘∅) ∈ ran 𝐺 |
20 | 16, 19 | eqeltrri 2910 | . . . 4 ⊢ 𝐶 ∈ ran 𝐺 |
21 | 7, 2 | om2uzsuci 13317 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
22 | oveq1 7163 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) + 1) = (𝑦 + 1)) | |
23 | 21, 22 | sylan9eq 2876 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) = (𝑦 + 1)) |
24 | peano2 7602 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω) | |
25 | fnfvelrn 6848 | . . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ suc 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺) | |
26 | 4, 24, 25 | sylancr 589 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) ∈ ran 𝐺) |
27 | 26 | adantr 483 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) ∈ ran 𝐺) |
28 | 23, 27 | eqeltrrd 2914 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝑦 + 1) ∈ ran 𝐺) |
29 | 28 | rexlimiva 3281 | . . . . . 6 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺) |
30 | 6, 29 | sylbi 219 | . . . . 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 12311 | . . 3 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → 𝑦 ∈ ran 𝐺) |
33 | 12, 32 | impbii 211 | . 2 ⊢ (𝑦 ∈ ran 𝐺 ↔ 𝑦 ∈ (ℤ≥‘𝐶)) |
34 | 33 | eqriv 2818 | 1 ⊢ ran 𝐺 = (ℤ≥‘𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ∅c0 4291 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 suc csuc 6193 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ωcom 7580 reccrdg 8045 1c1 10538 + caddc 10540 ℤcz 11982 ℤ≥cuz 12244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 |
This theorem is referenced by: om2uzf1oi 13322 |
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