Theorem om2uzsuc 4652

Description: The value of G (see om2uz0 4651) at a successor.

Hypotheses

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

Assertion

Ref	Expression
om2uzsuc	⊢ (A ∈ ω → (G ‘suc A) = ((G ‘A) + 1))

Distinct variable group(s):   x,y,C

Proof of Theorem om2uzsuc

Step	Hyp	Ref	Expression
1		suceq 2288	. . . 4 ⊢ (w = A → suc w = suc A)
2	1	fveq2d 2836	. . 3 ⊢ (w = A → (G ‘suc w) = (G ‘suc A))
3		fveq2 2832	. . . 4 ⊢ (w = A → (G ‘w) = (G ‘A))
4	3	opreq1d 3012	. . 3 ⊢ (w = A → ((G ‘w) + 1) = ((G ‘A) + 1))
5	2, 4	cleq12d 1115	. 2 ⊢ (w = A → ((G ‘suc w) = ((G ‘w) + 1) ↔ (G ‘suc A) = ((G ‘A) + 1)))
6		oprex 3018	. . 3 ⊢ ((G ‘w) + 1) ∈ V
7		ax-17 925	. . . 4 ⊢ (v ∈ C → ∀x v ∈ C)
8		ax-17 925	. . . 4 ⊢ (v ∈ w → ∀x v ∈ w)
9		hbopab1 2112	. . . . . . . . 9 ⊢ (v ∈ {⟨x, y⟩∣y = (x + 1)} → ∀x v ∈ {⟨x, y⟩∣y = (x + 1)})
10	9, 7	hbrdg 2974	. . . . . . . 8 ⊢ (v ∈ rec({⟨x, y⟩∣y = (x + 1)}, C) → ∀x v ∈ rec({⟨x, y⟩∣y = (x + 1)}, C))
11		ax-17 925	. . . . . . . 8 ⊢ (v ∈ ω → ∀x v ∈ ω)
12	10, 11	hbres 2577	. . . . . . 7 ⊢ (v ∈ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) → ∀x v ∈ (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω))
13	12, 8	hbfv 2837	. . . . . 6 ⊢ (v ∈ ((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w) → ∀x v ∈ ((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w))
14		ax-17 925	. . . . . 6 ⊢ (v ∈ + → ∀x v ∈ + )
15		ax-17 925	. . . . . 6 ⊢ (v ∈ 1 → ∀x v ∈ 1)
16	13, 14, 15	hbopr 3017	. . . . 5 ⊢ (v ∈ (((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w) + 1) → ∀x v ∈ (((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w) + 1))
17		om2uz.2	. . . . . . . 8 ⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)
18	17	fveq1i 2833	. . . . . . 7 ⊢ (G ‘w) = ((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w)
19	18	opreq1i 3009	. . . . . 6 ⊢ ((G ‘w) + 1) = (((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w) + 1)
20	19	eleq2i 1153	. . . . 5 ⊢ (v ∈ ((G ‘w) + 1) ↔ v ∈ (((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w) + 1))
21	20	bial 695	. . . . 5 ⊢ (∀x v ∈ ((G ‘w) + 1) ↔ ∀x v ∈ (((rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω) ‘w) + 1))
22	16, 20, 21	3imtr4 192	. . . 4 ⊢ (v ∈ ((G ‘w) + 1) → ∀x v ∈ ((G ‘w) + 1))
23		opreq1 3006	. . . 4 ⊢ (x = (G ‘w) → (x + 1) = ((G ‘w) + 1))
24	7, 8, 22, 17, 23	frsucopab 2992	. . 3 ⊢ ((w ∈ ω ∧ ((G ‘w) + 1) ∈ V) → (G ‘suc w) = ((G ‘w) + 1))
25	6, 24	mpan2 519	. 2 ⊢ (w ∈ ω → (G ‘suc w) = ((G ‘w) + 1))
26	5, 25	vtoclga 1387	1 ⊢ (A ∈ ω → (G ‘suc A) = ((G ‘A) + 1))

Syntax hints:   → wi 2  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {copab 2055  suc csuc 2201  ωcom 2372   ↾ cres 2412   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1c1 4029   + caddc 4031  ℤcz 4095

This theorem is referenced by:  om2uzuz 4653  om2uzlt 4654  om2uzran 4655  uzrdgsuc 4659

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

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-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  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-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-fv 2438  df-rdg 2970  df-opr 3003

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