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| Mirrors > Home > MPE Home > Th. List > om2uzrani | Structured version Visualization version GIF version | ||
| Description: Range of 𝐺 (see om2uz0i 13988). (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 8475 | . . . . . 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 6665 | . . . . . 6 ⊢ (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω) | 
| 4 | 1, 3 | mpbir 231 | . . . . 5 ⊢ 𝐺 Fn ω | 
| 5 | fvelrnb 6969 | . . . . 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 13990 | . . . . . 6 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ (ℤ≥‘𝐶)) | 
| 9 | eleq1 2829 | . . . . . 6 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) ↔ 𝑦 ∈ (ℤ≥‘𝐶))) | |
| 10 | 8, 9 | syl5ibcom 245 | . . . . 5 ⊢ (𝑧 ∈ ω → ((𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶))) | 
| 11 | 10 | rexlimiv 3148 | . . . 4 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → 𝑦 ∈ (ℤ≥‘𝐶)) | 
| 12 | 6, 11 | sylbi 217 | . . 3 ⊢ (𝑦 ∈ ran 𝐺 → 𝑦 ∈ (ℤ≥‘𝐶)) | 
| 13 | eleq1 2829 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ ran 𝐺 ↔ 𝐶 ∈ ran 𝐺)) | |
| 14 | eleq1 2829 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐺 ↔ 𝑦 ∈ ran 𝐺)) | |
| 15 | eleq1 2829 | . . . 4 ⊢ (𝑧 = (𝑦 + 1) → (𝑧 ∈ ran 𝐺 ↔ (𝑦 + 1) ∈ ran 𝐺)) | |
| 16 | 7, 2 | om2uz0i 13988 | . . . . 5 ⊢ (𝐺‘∅) = 𝐶 | 
| 17 | peano1 7910 | . . . . . 6 ⊢ ∅ ∈ ω | |
| 18 | fnfvelrn 7100 | . . . . . 6 ⊢ ((𝐺 Fn ω ∧ ∅ ∈ ω) → (𝐺‘∅) ∈ ran 𝐺) | |
| 19 | 4, 17, 18 | mp2an 692 | . . . . 5 ⊢ (𝐺‘∅) ∈ ran 𝐺 | 
| 20 | 16, 19 | eqeltrri 2838 | . . . 4 ⊢ 𝐶 ∈ ran 𝐺 | 
| 21 | 7, 2 | om2uzsuci 13989 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) | 
| 22 | oveq1 7438 | . . . . . . . . 9 ⊢ ((𝐺‘𝑧) = 𝑦 → ((𝐺‘𝑧) + 1) = (𝑦 + 1)) | |
| 23 | 21, 22 | sylan9eq 2797 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) = (𝑦 + 1)) | 
| 24 | peano2 7912 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ω → suc 𝑧 ∈ ω) | |
| 25 | fnfvelrn 7100 | . . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ suc 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺) | |
| 26 | 4, 24, 25 | sylancr 587 | . . . . . . . . 9 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) ∈ ran 𝐺) | 
| 27 | 26 | adantr 480 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝐺‘suc 𝑧) ∈ ran 𝐺) | 
| 28 | 23, 27 | eqeltrrd 2842 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ (𝐺‘𝑧) = 𝑦) → (𝑦 + 1) ∈ ran 𝐺) | 
| 29 | 28 | rexlimiva 3147 | . . . . . 6 ⊢ (∃𝑧 ∈ ω (𝐺‘𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺) | 
| 30 | 6, 29 | sylbi 217 | . . . . 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 12952 | . . 3 ⊢ (𝑦 ∈ (ℤ≥‘𝐶) → 𝑦 ∈ ran 𝐺) | 
| 33 | 12, 32 | impbii 209 | . 2 ⊢ (𝑦 ∈ ran 𝐺 ↔ 𝑦 ∈ (ℤ≥‘𝐶)) | 
| 34 | 33 | eqriv 2734 | 1 ⊢ ran 𝐺 = (ℤ≥‘𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 ran crn 5686 ↾ cres 5687 suc csuc 6386 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ωcom 7887 reccrdg 8449 1c1 11156 + caddc 11158 ℤcz 12613 ℤ≥cuz 12878 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 | 
| This theorem is referenced by: om2uzf1oi 13994 | 
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