Theorem om2uzran 4655

Description: Range of G (see om2uz0 4651).

Hypotheses

Ref	Expression
om2uz.1	⊢ C ∈ ℤ
om2uz.2	⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)

Assertion

Ref	Expression
om2uzran	⊢ ran G = {z ∈ ℤ∣C ≤ z}

Distinct variable group(s):   x,y,z   z,G   x,C,y,z

Proof of Theorem om2uzran

Step	Hyp	Ref	Expression
1		frfnom 2989	. . . . . 6 ⊢ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) Fn ω
2		om2uz.2	. . . . . . 7 ⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)
3		fneq1 2718	. . . . . . 7 ⊢ (G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) → (G Fn ω ↔ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) Fn ω))
4	2, 3	ax-mp 6	. . . . . 6 ⊢ (G Fn ω ↔ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) Fn ω)
5	1, 4	mpbir 165	. . . . 5 ⊢ G Fn ω
6		fvelrn 2883	. . . . 5 ⊢ (G Fn ω → (u ∈ ran G ↔ ∃w ∈ ω (G ‘w) = u))
7	5, 6	ax-mp 6	. . . 4 ⊢ (u ∈ ran G ↔ ∃w ∈ ω (G ‘w) = u)
8		eleq1 1149	. . . . . . 7 ⊢ ((G ‘w) = u → ((G ‘w) ∈ {z ∈ ℤ∣C ≤ z} ↔ u ∈ {z ∈ ℤ∣C ≤ z}))
9		om2uz.1	. . . . . . . 8 ⊢ C ∈ ℤ
10	9, 2	om2uzuz 4653	. . . . . . 7 ⊢ (w ∈ ω → (G ‘w) ∈ {z ∈ ℤ∣C ≤ z})
11	8, 10	syl5bi 183	. . . . . 6 ⊢ ((G ‘w) = u → (w ∈ ω → u ∈ {z ∈ ℤ∣C ≤ z}))
12	11	com12 13	. . . . 5 ⊢ (w ∈ ω → ((G ‘w) = u → u ∈ {z ∈ ℤ∣C ≤ z}))
13	12	r19.23aiv 1284	. . . 4 ⊢ (∃w ∈ ω (G ‘w) = u → u ∈ {z ∈ ℤ∣C ≤ z})
14	7, 13	sylbi 174	. . 3 ⊢ (u ∈ ran G → u ∈ {z ∈ ℤ∣C ≤ z})
15		breq2 2066	. . . . 5 ⊢ (z = u → (C ≤ z ↔ C ≤ u))
16	15	elrab 1422	. . . 4 ⊢ (u ∈ {z ∈ ℤ∣C ≤ z} ↔ (u ∈ ℤ ∧ C ≤ u))
17		eleq1 1149	. . . . . 6 ⊢ (w = C → (w ∈ ran G ↔ C ∈ ran G))
18		eleq1 1149	. . . . . 6 ⊢ (w = v → (w ∈ ran G ↔ v ∈ ran G))
19		eleq1 1149	. . . . . 6 ⊢ (w = (v + 1) → (w ∈ ran G ↔ (v + 1) ∈ ran G))
20		eleq1 1149	. . . . . 6 ⊢ (w = u → (w ∈ ran G ↔ u ∈ ran G))
21	9, 2	om2uz0 4651	. . . . . . 7 ⊢ (G ‘∅) = C
22		peano1 2390	. . . . . . . 8 ⊢ ∅ ∈ ω
23		fnfvrn 2889	. . . . . . . 8 ⊢ ((G Fn ω ∧ ∅ ∈ ω) → (G ‘∅) ∈ ran G)
24	5, 22, 23	mp2an 520	. . . . . . 7 ⊢ (G ‘∅) ∈ ran G
25	21, 24	eqeltrr 1160	. . . . . 6 ⊢ C ∈ ran G
26		fvelrn 2883	. . . . . . . . 9 ⊢ (G Fn ω → (v ∈ ran G ↔ ∃w ∈ ω (G ‘w) = v))
27	5, 26	ax-mp 6	. . . . . . . 8 ⊢ (v ∈ ran G ↔ ∃w ∈ ω (G ‘w) = v)
28	9, 2	om2uzsuc 4652	. . . . . . . . . . . 12 ⊢ (w ∈ ω → (G ‘suc w) = ((G ‘w) + 1))
29		opreq1 3006	. . . . . . . . . . . 12 ⊢ ((G ‘w) = v → ((G ‘w) + 1) = (v + 1))
30	28, 29	sylan9eq 1144	. . . . . . . . . . 11 ⊢ ((w ∈ ω ∧ (G ‘w) = v) → (G ‘suc w) = (v + 1))
31		peano2 2391	. . . . . . . . . . . . 13 ⊢ (w ∈ ω → suc w ∈ ω)
32		fnfvrn 2889	. . . . . . . . . . . . . 14 ⊢ ((G Fn ω ∧ suc w ∈ ω) → (G ‘suc w) ∈ ran G)
33	5, 32	mpan 518	. . . . . . . . . . . . 13 ⊢ (suc w ∈ ω → (G ‘suc w) ∈ ran G)
34	31, 33	syl 12	. . . . . . . . . . . 12 ⊢ (w ∈ ω → (G ‘suc w) ∈ ran G)
35	34	adantr 306	. . . . . . . . . . 11 ⊢ ((w ∈ ω ∧ (G ‘w) = v) → (G ‘suc w) ∈ ran G)
36	30, 35	eqeltrrd 1164	. . . . . . . . . 10 ⊢ ((w ∈ ω ∧ (G ‘w) = v) → (v + 1) ∈ ran G)
37	36	exp 291	. . . . . . . . 9 ⊢ (w ∈ ω → ((G ‘w) = v → (v + 1) ∈ ran G))
38	37	r19.23aiv 1284	. . . . . . . 8 ⊢ (∃w ∈ ω (G ‘w) = v → (v + 1) ∈ ran G)
39	27, 38	sylbi 174	. . . . . . 7 ⊢ (v ∈ ran G → (v + 1) ∈ ran G)
40	39	a1i 7	. . . . . 6 ⊢ (((v ∈ ℤ ∧ C ∈ ℤ) ∧ C ≤ v) → (v ∈ ran G → (v + 1) ∈ ran G))
41	17, 18, 19, 20, 25, 40	uzind 4603	. . . . 5 ⊢ (((u ∈ ℤ ∧ C ∈ ℤ) ∧ C ≤ u) → u ∈ ran G)
42	9, 41	mpan12 530	. . . 4 ⊢ ((u ∈ ℤ ∧ C ≤ u) → u ∈ ran G)
43	16, 42	sylbi 174	. . 3 ⊢ (u ∈ {z ∈ ℤ∣C ≤ z} → u ∈ ran G)
44	14, 43	impbi 139	. 2 ⊢ (u ∈ ran G ↔ u ∈ {z ∈ ℤ∣C ≤ z})
45	44	cleqri 1101	1 ⊢ ran G = {z ∈ ℤ∣C ≤ z}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  ∅c0 1707   class class class wbr 2054  {copab 2055  suc csuc 2201  ωcom 2372  ran crn 2411   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1c1 4029   + caddc 4031   ≤ cle 4092  ℤcz 4095

This theorem is referenced by:  om2uzf1o 4656  uzrdgval 4657

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535  df-z 4564

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