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Theorem om2uzrani 12734
 Description: Range of 𝐺 (see om2uz0i 12729). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.)
Hypotheses
Ref Expression
om2uz.1 𝐶 ∈ ℤ
om2uz.2 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
Assertion
Ref Expression
om2uzrani ran 𝐺 = (ℤ𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem om2uzrani
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frfnom 7515 . . . . . 6 (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω
2 om2uz.2 . . . . . . 7 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)
32fneq1i 5973 . . . . . 6 (𝐺 Fn ω ↔ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) Fn ω)
41, 3mpbir 221 . . . . 5 𝐺 Fn ω
5 fvelrnb 6230 . . . . 5 (𝐺 Fn ω → (𝑦 ∈ ran 𝐺 ↔ ∃𝑧 ∈ ω (𝐺𝑧) = 𝑦))
64, 5ax-mp 5 . . . 4 (𝑦 ∈ ran 𝐺 ↔ ∃𝑧 ∈ ω (𝐺𝑧) = 𝑦)
7 om2uz.1 . . . . . . 7 𝐶 ∈ ℤ
87, 2om2uzuzi 12731 . . . . . 6 (𝑧 ∈ ω → (𝐺𝑧) ∈ (ℤ𝐶))
9 eleq1 2687 . . . . . 6 ((𝐺𝑧) = 𝑦 → ((𝐺𝑧) ∈ (ℤ𝐶) ↔ 𝑦 ∈ (ℤ𝐶)))
108, 9syl5ibcom 235 . . . . 5 (𝑧 ∈ ω → ((𝐺𝑧) = 𝑦𝑦 ∈ (ℤ𝐶)))
1110rexlimiv 3023 . . . 4 (∃𝑧 ∈ ω (𝐺𝑧) = 𝑦𝑦 ∈ (ℤ𝐶))
126, 11sylbi 207 . . 3 (𝑦 ∈ ran 𝐺𝑦 ∈ (ℤ𝐶))
13 eleq1 2687 . . . 4 (𝑧 = 𝐶 → (𝑧 ∈ ran 𝐺𝐶 ∈ ran 𝐺))
14 eleq1 2687 . . . 4 (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐺))
15 eleq1 2687 . . . 4 (𝑧 = (𝑦 + 1) → (𝑧 ∈ ran 𝐺 ↔ (𝑦 + 1) ∈ ran 𝐺))
167, 2om2uz0i 12729 . . . . 5 (𝐺‘∅) = 𝐶
17 peano1 7070 . . . . . 6 ∅ ∈ ω
18 fnfvelrn 6342 . . . . . 6 ((𝐺 Fn ω ∧ ∅ ∈ ω) → (𝐺‘∅) ∈ ran 𝐺)
194, 17, 18mp2an 707 . . . . 5 (𝐺‘∅) ∈ ran 𝐺
2016, 19eqeltrri 2696 . . . 4 𝐶 ∈ ran 𝐺
217, 2om2uzsuci 12730 . . . . . . . . 9 (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺𝑧) + 1))
22 oveq1 6642 . . . . . . . . 9 ((𝐺𝑧) = 𝑦 → ((𝐺𝑧) + 1) = (𝑦 + 1))
2321, 22sylan9eq 2674 . . . . . . . 8 ((𝑧 ∈ ω ∧ (𝐺𝑧) = 𝑦) → (𝐺‘suc 𝑧) = (𝑦 + 1))
24 peano2 7071 . . . . . . . . . 10 (𝑧 ∈ ω → suc 𝑧 ∈ ω)
25 fnfvelrn 6342 . . . . . . . . . 10 ((𝐺 Fn ω ∧ suc 𝑧 ∈ ω) → (𝐺‘suc 𝑧) ∈ ran 𝐺)
264, 24, 25sylancr 694 . . . . . . . . 9 (𝑧 ∈ ω → (𝐺‘suc 𝑧) ∈ ran 𝐺)
2726adantr 481 . . . . . . . 8 ((𝑧 ∈ ω ∧ (𝐺𝑧) = 𝑦) → (𝐺‘suc 𝑧) ∈ ran 𝐺)
2823, 27eqeltrrd 2700 . . . . . . 7 ((𝑧 ∈ ω ∧ (𝐺𝑧) = 𝑦) → (𝑦 + 1) ∈ ran 𝐺)
2928rexlimiva 3024 . . . . . 6 (∃𝑧 ∈ ω (𝐺𝑧) = 𝑦 → (𝑦 + 1) ∈ ran 𝐺)
306, 29sylbi 207 . . . . 5 (𝑦 ∈ ran 𝐺 → (𝑦 + 1) ∈ ran 𝐺)
3130a1i 11 . . . 4 (𝑦 ∈ (ℤ𝐶) → (𝑦 ∈ ran 𝐺 → (𝑦 + 1) ∈ ran 𝐺))
327, 13, 14, 15, 14, 20, 31uzind4i 11735 . . 3 (𝑦 ∈ (ℤ𝐶) → 𝑦 ∈ ran 𝐺)
3312, 32impbii 199 . 2 (𝑦 ∈ ran 𝐺𝑦 ∈ (ℤ𝐶))
3433eqriv 2617 1 ran 𝐺 = (ℤ𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∃wrex 2910  Vcvv 3195  ∅c0 3907   ↦ cmpt 4720  ran crn 5105   ↾ cres 5106  suc csuc 5713   Fn wfn 5871  ‘cfv 5876  (class class class)co 6635  ωcom 7050  reccrdg 7490  1c1 9922   + caddc 9924  ℤcz 11362  ℤ≥cuz 11672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673 This theorem is referenced by:  om2uzf1oi  12735
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