Theorem frec2uzrdg 8876

Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ₀) with characteristic function 𝐹(x, y) and initial value A. This lemma shows that evaluating 𝑅 at an element of 𝜔 gives an ordered pair whose first element is the index (translated from 𝜔 to (ℤ_≥‘𝐶)). See comment in frec2uz0d 8866 which describes 𝐺 and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	⊢ (φ → 𝐶 ∈ ℤ)
frec2uz.2	⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶)
uzrdg.s	⊢ (φ → 𝑆 ∈ 𝑉)
uzrdg.a	⊢ (φ → A ∈ 𝑆)
uzrdg.f	⊢ ((φ ∧ (x ∈ (ℤ≥‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆)
uzrdg.2	⊢ 𝑅 = frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)
uzrdg.b	⊢ (φ → B ∈ 𝜔)

Assertion

Ref	Expression
frec2uzrdg	⊢ (φ → (𝑅‘B) = ⟨(𝐺‘B), (2nd ‘(𝑅‘B))⟩)

Distinct variable groups:   y,A   x,𝐶,y   y,𝐺   x,𝐹,y   x,𝑆,y   φ,x,y

Allowed substitution hints:   A(x)   B(x,y)   𝑅(x,y)   𝐺(x)   𝑉(x,y)

Proof of Theorem frec2uzrdg

Dummy variables w z v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzrdg.b	. 2 ⊢ (φ → B ∈ 𝜔)
2		fveq2 5121	. . . . 5 ⊢ (z = B → (𝑅‘z) = (𝑅‘B))
3		fveq2 5121	. . . . . 6 ⊢ (z = B → (𝐺‘z) = (𝐺‘B))
4	2	fveq2d 5125	. . . . . 6 ⊢ (z = B → (2nd ‘(𝑅‘z)) = (2nd ‘(𝑅‘B)))
5	3, 4	opeq12d 3548	. . . . 5 ⊢ (z = B → ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ = ⟨(𝐺‘B), (2nd ‘(𝑅‘B))⟩)
6	2, 5	eqeq12d 2051	. . . 4 ⊢ (z = B → ((𝑅‘z) = ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ ↔ (𝑅‘B) = ⟨(𝐺‘B), (2nd ‘(𝑅‘B))⟩))
7	6	imbi2d 219	. . 3 ⊢ (z = B → ((φ → (𝑅‘z) = ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩) ↔ (φ → (𝑅‘B) = ⟨(𝐺‘B), (2nd ‘(𝑅‘B))⟩)))
8		fveq2 5121	. . . . 5 ⊢ (z = ∅ → (𝑅‘z) = (𝑅‘∅))
9		fveq2 5121	. . . . . 6 ⊢ (z = ∅ → (𝐺‘z) = (𝐺‘∅))
10	8	fveq2d 5125	. . . . . 6 ⊢ (z = ∅ → (2nd ‘(𝑅‘z)) = (2nd ‘(𝑅‘∅)))
11	9, 10	opeq12d 3548	. . . . 5 ⊢ (z = ∅ → ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
12	8, 11	eqeq12d 2051	. . . 4 ⊢ (z = ∅ → ((𝑅‘z) = ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ ↔ (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩))
13		fveq2 5121	. . . . 5 ⊢ (z = v → (𝑅‘z) = (𝑅‘v))
14		fveq2 5121	. . . . . 6 ⊢ (z = v → (𝐺‘z) = (𝐺‘v))
15	13	fveq2d 5125	. . . . . 6 ⊢ (z = v → (2nd ‘(𝑅‘z)) = (2nd ‘(𝑅‘v)))
16	14, 15	opeq12d 3548	. . . . 5 ⊢ (z = v → ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩)
17	13, 16	eqeq12d 2051	. . . 4 ⊢ (z = v → ((𝑅‘z) = ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ ↔ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩))
18		fveq2 5121	. . . . 5 ⊢ (z = suc v → (𝑅‘z) = (𝑅‘suc v))
19		fveq2 5121	. . . . . 6 ⊢ (z = suc v → (𝐺‘z) = (𝐺‘suc v))
20	18	fveq2d 5125	. . . . . 6 ⊢ (z = suc v → (2nd ‘(𝑅‘z)) = (2nd ‘(𝑅‘suc v)))
21	19, 20	opeq12d 3548	. . . . 5 ⊢ (z = suc v → ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩)
22	18, 21	eqeq12d 2051	. . . 4 ⊢ (z = suc v → ((𝑅‘z) = ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩ ↔ (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩))
23		uzrdg.2	. . . . . . 7 ⊢ 𝑅 = frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)
24	23	fveq1i 5122	. . . . . 6 ⊢ (𝑅‘∅) = (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘∅)
25		frec2uz.1	. . . . . . . 8 ⊢ (φ → 𝐶 ∈ ℤ)
26		uzrdg.a	. . . . . . . 8 ⊢ (φ → A ∈ 𝑆)
27		opexg 3955	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ A ∈ 𝑆) → ⟨𝐶, A⟩ ∈ V)
28	25, 26, 27	syl2anc 391	. . . . . . 7 ⊢ (φ → ⟨𝐶, A⟩ ∈ V)
29		frec0g 5922	. . . . . . 7 ⊢ (⟨𝐶, A⟩ ∈ V → (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘∅) = ⟨𝐶, A⟩)
30	28, 29	syl 14	. . . . . 6 ⊢ (φ → (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘∅) = ⟨𝐶, A⟩)
31	24, 30	syl5eq 2081	. . . . 5 ⊢ (φ → (𝑅‘∅) = ⟨𝐶, A⟩)
32		frec2uz.2	. . . . . . 7 ⊢ 𝐺 = frec((x ∈ ℤ ↦ (x + 1)), 𝐶)
33	25, 32	frec2uz0d 8866	. . . . . 6 ⊢ (φ → (𝐺‘∅) = 𝐶)
34	31	fveq2d 5125	. . . . . . 7 ⊢ (φ → (2nd ‘(𝑅‘∅)) = (2nd ‘⟨𝐶, A⟩))
35		uzid 8263	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
36	25, 35	syl 14	. . . . . . . 8 ⊢ (φ → 𝐶 ∈ (ℤ≥‘𝐶))
37		op2ndg 5720	. . . . . . . 8 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ A ∈ 𝑆) → (2nd ‘⟨𝐶, A⟩) = A)
38	36, 26, 37	syl2anc 391	. . . . . . 7 ⊢ (φ → (2nd ‘⟨𝐶, A⟩) = A)
39	34, 38	eqtrd 2069	. . . . . 6 ⊢ (φ → (2nd ‘(𝑅‘∅)) = A)
40	33, 39	opeq12d 3548	. . . . 5 ⊢ (φ → ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩ = ⟨𝐶, A⟩)
41	31, 40	eqtr4d 2072	. . . 4 ⊢ (φ → (𝑅‘∅) = ⟨(𝐺‘∅), (2nd ‘(𝑅‘∅))⟩)
42		zex 8030	. . . . . . . . . . . . . . . 16 ⊢ ℤ ∈ V
43		uzssz 8268	. . . . . . . . . . . . . . . 16 ⊢ (ℤ≥‘𝐶) ⊆ ℤ
44	42, 43	ssexi 3886	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝐶) ∈ V
45	44	a1i 9	. . . . . . . . . . . . . 14 ⊢ ((φ ∧ v ∈ 𝜔) → (ℤ≥‘𝐶) ∈ V)
46		uzrdg.s	. . . . . . . . . . . . . . 15 ⊢ (φ → 𝑆 ∈ 𝑉)
47	46	adantr 261	. . . . . . . . . . . . . 14 ⊢ ((φ ∧ v ∈ 𝜔) → 𝑆 ∈ 𝑉)
48		mpt2exga 5777	. . . . . . . . . . . . . 14 ⊢ (((ℤ≥‘𝐶) ∈ V ∧ 𝑆 ∈ 𝑉) → (x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) ∈ V)
49	45, 47, 48	syl2anc 391	. . . . . . . . . . . . 13 ⊢ ((φ ∧ v ∈ 𝜔) → (x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) ∈ V)
50		vex 2554	. . . . . . . . . . . . . 14 ⊢ z ∈ V
51	50	a1i 9	. . . . . . . . . . . . 13 ⊢ ((φ ∧ v ∈ 𝜔) → z ∈ V)
52		fvexg 5137	. . . . . . . . . . . . 13 ⊢ (((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) ∈ V ∧ z ∈ V) → ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) ∈ V)
53	49, 51, 52	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ 𝜔) → ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) ∈ V)
54	53	alrimiv 1751	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → ∀z((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) ∈ V)
55	28	adantr 261	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → ⟨𝐶, A⟩ ∈ V)
56		simpr 103	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → v ∈ 𝜔)
57		frecsuc 5930	. . . . . . . . . . 11 ⊢ ((∀z((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘z) ∈ V ∧ ⟨𝐶, A⟩ ∈ V ∧ v ∈ 𝜔) → (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘suc v) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v)))
58	54, 55, 56, 57	syl3anc 1134	. . . . . . . . . 10 ⊢ ((φ ∧ v ∈ 𝜔) → (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘suc v) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v)))
59	23	fveq1i 5122	. . . . . . . . . 10 ⊢ (𝑅‘suc v) = (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘suc v)
60	23	fveq1i 5122	. . . . . . . . . . 11 ⊢ (𝑅‘v) = (frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v)
61	60	fveq2i 5124	. . . . . . . . . 10 ⊢ ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅‘v)) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(frec((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩), ⟨𝐶, A⟩)‘v))
62	58, 59, 61	3eqtr4g 2094	. . . . . . . . 9 ⊢ ((φ ∧ v ∈ 𝜔) → (𝑅‘suc v) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅‘v)))
63	62	adantr 261	. . . . . . . 8 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (𝑅‘suc v) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅‘v)))
64		fveq2 5121	. . . . . . . . 9 ⊢ ((𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩ → ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅‘v)) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩))
65		df-ov 5458	. . . . . . . . . 10 ⊢ ((𝐺‘v)(x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)(2nd ‘(𝑅‘v))) = ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩)
66	25	adantr 261	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ 𝜔) → 𝐶 ∈ ℤ)
67	66, 32, 56	frec2uzuzd 8869	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → (𝐺‘v) ∈ (ℤ≥‘𝐶))
68		uzrdg.f	. . . . . . . . . . . . 13 ⊢ ((φ ∧ (x ∈ (ℤ≥‘𝐶) ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆)
69	25, 32, 46, 26, 68, 23	frecuzrdgrrn 8875	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ 𝜔) → (𝑅‘v) ∈ ((ℤ≥‘𝐶) × 𝑆))
70		xp2nd 5735	. . . . . . . . . . . 12 ⊢ ((𝑅‘v) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘v)) ∈ 𝑆)
71	69, 70	syl 14	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → (2nd ‘(𝑅‘v)) ∈ 𝑆)
72		peano2uz 8302	. . . . . . . . . . . . 13 ⊢ ((𝐺‘v) ∈ (ℤ≥‘𝐶) → ((𝐺‘v) + 1) ∈ (ℤ≥‘𝐶))
73	67, 72	syl 14	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ 𝜔) → ((𝐺‘v) + 1) ∈ (ℤ≥‘𝐶))
74	68	caovclg 5595	. . . . . . . . . . . . . 14 ⊢ ((φ ∧ (z ∈ (ℤ≥‘𝐶) ∧ w ∈ 𝑆)) → (z𝐹w) ∈ 𝑆)
75	74	adantlr 446	. . . . . . . . . . . . 13 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (z ∈ (ℤ≥‘𝐶) ∧ w ∈ 𝑆)) → (z𝐹w) ∈ 𝑆)
76	75, 67, 71	caovcld 5596	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ 𝜔) → ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))) ∈ 𝑆)
77		opelxp 4317	. . . . . . . . . . . 12 ⊢ (⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((𝐺‘v) + 1) ∈ (ℤ≥‘𝐶) ∧ ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))) ∈ 𝑆))
78	73, 76, 77	sylanbrc 394	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
79		oveq1 5462	. . . . . . . . . . . . 13 ⊢ (w = (𝐺‘v) → (w + 1) = ((𝐺‘v) + 1))
80		oveq1 5462	. . . . . . . . . . . . 13 ⊢ (w = (𝐺‘v) → (w𝐹z) = ((𝐺‘v)𝐹z))
81	79, 80	opeq12d 3548	. . . . . . . . . . . 12 ⊢ (w = (𝐺‘v) → ⟨(w + 1), (w𝐹z)⟩ = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹z)⟩)
82		oveq2 5463	. . . . . . . . . . . . 13 ⊢ (z = (2nd ‘(𝑅‘v)) → ((𝐺‘v)𝐹z) = ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))))
83	82	opeq2d 3547	. . . . . . . . . . . 12 ⊢ (z = (2nd ‘(𝑅‘v)) → ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹z)⟩ = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
84		oveq1 5462	. . . . . . . . . . . . . 14 ⊢ (x = w → (x + 1) = (w + 1))
85		oveq1 5462	. . . . . . . . . . . . . 14 ⊢ (x = w → (x𝐹y) = (w𝐹y))
86	84, 85	opeq12d 3548	. . . . . . . . . . . . 13 ⊢ (x = w → ⟨(x + 1), (x𝐹y)⟩ = ⟨(w + 1), (w𝐹y)⟩)
87		oveq2 5463	. . . . . . . . . . . . . 14 ⊢ (y = z → (w𝐹y) = (w𝐹z))
88	87	opeq2d 3547	. . . . . . . . . . . . 13 ⊢ (y = z → ⟨(w + 1), (w𝐹y)⟩ = ⟨(w + 1), (w𝐹z)⟩)
89	86, 88	cbvmpt2v 5526	. . . . . . . . . . . 12 ⊢ (x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩) = (w ∈ (ℤ≥‘𝐶), z ∈ 𝑆 ↦ ⟨(w + 1), (w𝐹z)⟩)
90	81, 83, 89	ovmpt2g 5577	. . . . . . . . . . 11 ⊢ (((𝐺‘v) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘v)) ∈ 𝑆 ∧ ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝐺‘v)(x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)(2nd ‘(𝑅‘v))) = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
91	67, 71, 78, 90	syl3anc 1134	. . . . . . . . . 10 ⊢ ((φ ∧ v ∈ 𝜔) → ((𝐺‘v)(x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)(2nd ‘(𝑅‘v))) = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
92	65, 91	syl5eqr 2083	. . . . . . . . 9 ⊢ ((φ ∧ v ∈ 𝜔) → ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
93	64, 92	sylan9eqr 2091	. . . . . . . 8 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → ((x ∈ (ℤ≥‘𝐶), y ∈ 𝑆 ↦ ⟨(x + 1), (x𝐹y)⟩)‘(𝑅‘v)) = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
94	63, 93	eqtrd 2069	. . . . . . 7 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (𝑅‘suc v) = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
95	66, 32, 56	frec2uzsucd 8868	. . . . . . . . 9 ⊢ ((φ ∧ v ∈ 𝜔) → (𝐺‘suc v) = ((𝐺‘v) + 1))
96	95	adantr 261	. . . . . . . 8 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (𝐺‘suc v) = ((𝐺‘v) + 1))
97	94	fveq2d 5125	. . . . . . . . 9 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (2nd ‘(𝑅‘suc v)) = (2nd ‘⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩))
98	66, 32, 56	frec2uzzd 8867	. . . . . . . . . . . 12 ⊢ ((φ ∧ v ∈ 𝜔) → (𝐺‘v) ∈ ℤ)
99	98	peano2zd 8139	. . . . . . . . . . 11 ⊢ ((φ ∧ v ∈ 𝜔) → ((𝐺‘v) + 1) ∈ ℤ)
100	99	adantr 261	. . . . . . . . . 10 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → ((𝐺‘v) + 1) ∈ ℤ)
101	76	adantr 261	. . . . . . . . . 10 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))) ∈ 𝑆)
102		op2ndg 5720	. . . . . . . . . 10 ⊢ ((((𝐺‘v) + 1) ∈ ℤ ∧ ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))) ∈ 𝑆) → (2nd ‘⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩) = ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))))
103	100, 101, 102	syl2anc 391	. . . . . . . . 9 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (2nd ‘⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩) = ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))))
104	97, 103	eqtrd 2069	. . . . . . . 8 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (2nd ‘(𝑅‘suc v)) = ((𝐺‘v)𝐹(2nd ‘(𝑅‘v))))
105	96, 104	opeq12d 3548	. . . . . . 7 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩ = ⟨((𝐺‘v) + 1), ((𝐺‘v)𝐹(2nd ‘(𝑅‘v)))⟩)
106	94, 105	eqtr4d 2072	. . . . . 6 ⊢ (((φ ∧ v ∈ 𝜔) ∧ (𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩) → (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩)
107	106	ex 108	. . . . 5 ⊢ ((φ ∧ v ∈ 𝜔) → ((𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩ → (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩))
108	107	expcom 109	. . . 4 ⊢ (v ∈ 𝜔 → (φ → ((𝑅‘v) = ⟨(𝐺‘v), (2nd ‘(𝑅‘v))⟩ → (𝑅‘suc v) = ⟨(𝐺‘suc v), (2nd ‘(𝑅‘suc v))⟩)))
109	12, 17, 22, 41, 108	finds2 4267	. . 3 ⊢ (z ∈ 𝜔 → (φ → (𝑅‘z) = ⟨(𝐺‘z), (2nd ‘(𝑅‘z))⟩))
110	7, 109	vtoclga 2613	. 2 ⊢ (B ∈ 𝜔 → (φ → (𝑅‘B) = ⟨(𝐺‘B), (2nd ‘(𝑅‘B))⟩))
111	1, 110	mpcom 32	1 ⊢ (φ → (𝑅‘B) = ⟨(𝐺‘B), (2nd ‘(𝑅‘B))⟩)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ∅c0 3218  ⟨cop 3370   ↦ cmpt 3809  suc csuc 4068  𝜔com 4256   × cxp 4286  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  2^nd c2nd 5708  freccfrec 5917  1c1 6712   + caddc 6714  ℤcz 8021  ℤ_≥cuz 8249

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-ltadd 6799

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250

This theorem is referenced by:  frecuzrdglem  8878  frecuzrdgfn  8879  frecuzrdgsuc  8882