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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  cz 11362  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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