Theorem seq3val 10649

Description: Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10653, seq3-1 10651 and seq3p1 10654, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)

Hypotheses

Ref	Expression
seq3val.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
seq3val.r	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
seq3val.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
seq3val.pl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
seq3val	⊢ (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)

Distinct variable groups:   𝑥, + ,𝑦,𝑤,𝑧   𝑥,𝐹,𝑦,𝑤,𝑧   𝑥,𝑀,𝑦,𝑤,𝑧   𝑥,𝑅,𝑦,𝑤,𝑧   𝑥,𝑆,𝑦,𝑤,𝑧   𝜑,𝑥,𝑦,𝑤,𝑧

Proof of Theorem seq3val

Dummy variables 𝑎 𝑏 𝑘 𝑐 𝑛 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-seqfrec 10637	. 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
2		seq3val.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		fveq2 5603	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
4	3	eleq1d 2278	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆))
5		seq3val.f	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
6	5	ralrimiva 2583	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
7		uzid 9704	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
8	2, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
9	4, 6, 8	rspcdva 2892	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆)
10		ssv 3226	. . . . . . 7 ⊢ 𝑆 ⊆ V
11	10	a1i 9	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ V)
12		seq3val.pl	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
13	5, 12	iseqovex 10647	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)
14		seq3val.r	. . . . . 6 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
15	2, 9, 11, 13, 14	frecuzrdgrclt 10604	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆))
16		ffn 5449	. . . . 5 ⊢ (𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆) → 𝑅 Fn ω)
17	15, 16	syl 14	. . . 4 ⊢ (𝜑 → 𝑅 Fn ω)
18		1st2nd2 6291	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
19	18	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → 𝑢 = ⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
20	19	fveq2d 5607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
21		df-ov 5977	. . . . . . . . . 10 ⊢ ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘𝑢)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
22	20, 21	eqtr4di 2260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘𝑢)))
23		xp1st 6281	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆) → (1st ‘𝑢) ∈ (ℤ≥‘𝑀))
24	23	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (1st ‘𝑢) ∈ (ℤ≥‘𝑀))
25		xp2nd 6282	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆) → (2nd ‘𝑢) ∈ 𝑆)
26	25	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (2nd ‘𝑢) ∈ 𝑆)
27	26	elexd 2793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (2nd ‘𝑢) ∈ V)
28		peano2uz 9746	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑢) ∈ (ℤ≥‘𝑀) → ((1st ‘𝑢) + 1) ∈ (ℤ≥‘𝑀))
29	24, 28	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢) + 1) ∈ (ℤ≥‘𝑀))
30	12	caovclg 6129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
31	30	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
32		fveq2 5603	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((1st ‘𝑢) + 1) → (𝐹‘𝑥) = (𝐹‘((1st ‘𝑢) + 1)))
33	32	eleq1d 2278	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((1st ‘𝑢) + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘((1st ‘𝑢) + 1)) ∈ 𝑆))
34	6	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
35	33, 34, 29	rspcdva 2892	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (𝐹‘((1st ‘𝑢) + 1)) ∈ 𝑆)
36	31, 26, 35	caovcld 6130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))) ∈ 𝑆)
37		opelxpi 4728	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑢) + 1) ∈ (ℤ≥‘𝑀) ∧ ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))) ∈ 𝑆) → ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
38	29, 36, 37	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
39		oveq1 5981	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑢) → (𝑥 + 1) = ((1st ‘𝑢) + 1))
40		fvoveq1 5997	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑢) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘𝑢) + 1)))
41	40	oveq2d 5990	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑢) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘𝑢) + 1))))
42	39, 41	opeq12d 3844	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑢) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘𝑢) + 1), (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))⟩)
43		oveq1 5981	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑢) → (𝑦 + (𝐹‘((1st ‘𝑢) + 1))) = ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))))
44	43	opeq2d 3843	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑢) → ⟨((1st ‘𝑢) + 1), (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))⟩ = ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩)
45		eqid 2209	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
46	42, 44, 45	ovmpog 6110	. . . . . . . . . 10 ⊢ (((1st ‘𝑢) ∈ (ℤ≥‘𝑀) ∧ (2nd ‘𝑢) ∈ V ∧ ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘𝑢)) = ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩)
47	24, 27, 38, 46	syl3anc 1252	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘𝑢)) = ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩)
48	22, 47	eqtrd 2242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) = ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩)
49	48, 38	eqeltrd 2286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
50	49	ralrimiva 2583	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
51		opelxpi 4728	. . . . . . 7 ⊢ ((𝑀 ∈ (ℤ≥‘𝑀) ∧ (𝐹‘𝑀) ∈ 𝑆) → ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
52	8, 9, 51	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
53	50, 52	jca 306	. . . . 5 ⊢ (𝜑 → (∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆)))
54		frecfcl 6521	. . . . 5 ⊢ ((∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆)) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩):ω⟶((ℤ≥‘𝑀) × 𝑆))
55		ffn 5449	. . . . 5 ⊢ (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩):ω⟶((ℤ≥‘𝑀) × 𝑆) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) Fn ω)
56	53, 54, 55	3syl 17	. . . 4 ⊢ (𝜑 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) Fn ω)
57		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = ∅ → (𝑅‘𝑐) = (𝑅‘∅))
58		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = ∅ → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅))
59	57, 58	eqeq12d 2224	. . . . . . 7 ⊢ (𝑐 = ∅ → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) ↔ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅)))
60	59	imbi2d 230	. . . . . 6 ⊢ (𝑐 = ∅ → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅))))
61		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = 𝑘 → (𝑅‘𝑐) = (𝑅‘𝑘))
62		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = 𝑘 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘))
63	61, 62	eqeq12d 2224	. . . . . . 7 ⊢ (𝑐 = 𝑘 → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) ↔ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
64	63	imbi2d 230	. . . . . 6 ⊢ (𝑐 = 𝑘 → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘))))
65		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = suc 𝑘 → (𝑅‘𝑐) = (𝑅‘suc 𝑘))
66		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = suc 𝑘 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘))
67	65, 66	eqeq12d 2224	. . . . . . 7 ⊢ (𝑐 = suc 𝑘 → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) ↔ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘)))
68	67	imbi2d 230	. . . . . 6 ⊢ (𝑐 = suc 𝑘 → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘))))
69		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = 𝑛 → (𝑅‘𝑐) = (𝑅‘𝑛))
70		fveq2 5603	. . . . . . . 8 ⊢ (𝑐 = 𝑛 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑛))
71	69, 70	eqeq12d 2224	. . . . . . 7 ⊢ (𝑐 = 𝑛 → ((𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐) ↔ (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑛)))
72	71	imbi2d 230	. . . . . 6 ⊢ (𝑐 = 𝑛 → ((𝜑 → (𝑅‘𝑐) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑐)) ↔ (𝜑 → (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑛))))
73	14	fveq1i 5604	. . . . . . . 8 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅)
74		frec0g 6513	. . . . . . . . 9 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅) = ⟨𝑀, (𝐹‘𝑀)⟩)
75	52, 74	syl 14	. . . . . . . 8 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅) = ⟨𝑀, (𝐹‘𝑀)⟩)
76	73, 75	eqtrid 2254	. . . . . . 7 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝑀, (𝐹‘𝑀)⟩)
77		frec0g 6513	. . . . . . . 8 ⊢ (⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅) = ⟨𝑀, (𝐹‘𝑀)⟩)
78	52, 77	syl 14	. . . . . . 7 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅) = ⟨𝑀, (𝐹‘𝑀)⟩)
79	76, 78	eqtr4d 2245	. . . . . 6 ⊢ (𝜑 → (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘∅))
80	15	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → 𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆))
81		simpll 527	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → 𝑘 ∈ ω)
82	80, 81	ffvelcdmd 5744	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆))
83		xp1st 6281	. . . . . . . . . . 11 ⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆) → (1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀))
84	82, 83	syl 14	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀))
85		xp2nd 6282	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑆)
86	82, 85	syl 14	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑆)
87	86	elexd 2793	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ V)
88	30	adantll 476	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
89	88	adantlr 477	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 + 𝑏) ∈ 𝑆)
90		fveq2 5603	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ((1st ‘(𝑅‘𝑘)) + 1) → (𝐹‘𝑎) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))
91	90	eleq1d 2278	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ((1st ‘(𝑅‘𝑘)) + 1) → ((𝐹‘𝑎) ∈ 𝑆 ↔ (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)) ∈ 𝑆))
92		fveq2 5603	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
93	92	eleq1d 2278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑎) ∈ 𝑆))
94	93	cbvralv 2745	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆 ↔ ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐹‘𝑎) ∈ 𝑆)
95	6, 94	sylib 122	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐹‘𝑎) ∈ 𝑆)
96	95	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐹‘𝑎) ∈ 𝑆)
97		peano2uz 9746	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) → ((1st ‘(𝑅‘𝑘)) + 1) ∈ (ℤ≥‘𝑀))
98	84, 97	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅‘𝑘)) + 1) ∈ (ℤ≥‘𝑀))
99	91, 96, 98	rspcdva 2892	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)) ∈ 𝑆)
100	89, 86, 99	caovcld 6130	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) ∈ 𝑆)
101		fvoveq1 5997	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1st ‘(𝑅‘𝑘)) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))
102	101	oveq2d 5990	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘(𝑅‘𝑘)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
103		oveq1 5981	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (2nd ‘(𝑅‘𝑘)) → (𝑤 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
104		eqid 2209	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
105	102, 103, 104	ovmpog 6110	. . . . . . . . . . . . 13 ⊢ (((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) ∧ (2nd ‘(𝑅‘𝑘)) ∈ 𝑆 ∧ ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) ∈ 𝑆) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
106	84, 86, 100, 105	syl3anc 1252	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
107	106	opeq2d 3843	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩ = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
108	106, 100	eqeltrd 2286	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) ∈ 𝑆)
109		opelxpi 4728	. . . . . . . . . . . 12 ⊢ ((((1st ‘(𝑅‘𝑘)) + 1) ∈ (ℤ≥‘𝑀) ∧ ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))) ∈ 𝑆) → ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
110	98, 108, 109	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
111	107, 110	eqeltrrd 2287	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
112		oveq1 5981	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑥 + 1) = ((1st ‘(𝑅‘𝑘)) + 1))
113		fvoveq1 5997	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))
114	113	oveq2d 5990	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
115	112, 114	opeq12d 3844	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘(𝑅‘𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
116		oveq1 5981	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
117	116	opeq2d 3843	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ⟨((1st ‘(𝑅‘𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩ = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
118	115, 117, 45	ovmpog 6110	. . . . . . . . . 10 ⊢ (((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) ∧ (2nd ‘(𝑅‘𝑘)) ∈ V ∧ ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅‘𝑘))) = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
119	84, 87, 111, 118	syl3anc 1252	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅‘𝑘))) = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
120	50	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
121	52	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
122		frecsuc 6523	. . . . . . . . . . . 12 ⊢ ((∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
123	120, 121, 81, 122	syl3anc 1252	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
124		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘))
125	124	fveq2d 5607	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
126	123, 125	eqtr4d 2245	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅‘𝑘)))
127		1st2nd2 6291	. . . . . . . . . . . . 13 ⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝑀) × 𝑆) → (𝑅‘𝑘) = ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩)
128	82, 127	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘𝑘) = ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩)
129	128	fveq2d 5607	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩))
130		df-ov 5977	. . . . . . . . . . 11 ⊢ ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅‘𝑘))) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩)
131	129, 130	eqtr4di 2260	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘(𝑅‘𝑘)) = ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅‘𝑘))))
132	126, 131	eqtrd 2242	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘) = ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)(2nd ‘(𝑅‘𝑘))))
133	14	fveq1i 5604	. . . . . . . . . . . . . . 15 ⊢ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘)
134	19	fveq2d 5607	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩))
135		df-ov 5977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘𝑢)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘𝑢), (2nd ‘𝑢)⟩)
136	134, 135	eqtr4di 2260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘𝑢)))
137		fvoveq1 5997	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = (1st ‘𝑢) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘𝑢) + 1)))
138	137	oveq2d 5990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (1st ‘𝑢) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘𝑢) + 1))))
139		oveq1 5981	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (2nd ‘𝑢) → (𝑤 + (𝐹‘((1st ‘𝑢) + 1))) = ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))))
140	138, 139, 104	ovmpog 6110	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1st ‘𝑢) ∈ (ℤ≥‘𝑀) ∧ (2nd ‘𝑢) ∈ 𝑆 ∧ ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))) ∈ 𝑆) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) = ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))))
141	24, 26, 36, 140	syl3anc 1252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) = ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))))
142	141, 36	eqeltrd 2286	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) ∈ 𝑆)
143		opelxpi 4728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((1st ‘𝑢) + 1) ∈ (ℤ≥‘𝑀) ∧ ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) ∈ 𝑆) → ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
144	29, 142, 143	syl2anc 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆))
145		oveq1 5981	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = (1st ‘𝑢) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
146	39, 145	opeq12d 3844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (1st ‘𝑢) → ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
147		oveq2 5982	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (2nd ‘𝑢) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)))
148	147	opeq2d 3843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (2nd ‘𝑢) → ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩)
149		eqid 2209	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
150	146, 148, 149	ovmpog 6110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑢) ∈ (ℤ≥‘𝑀) ∧ (2nd ‘𝑢) ∈ V ∧ ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘𝑢)) = ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩)
151	24, 27, 144, 150	syl3anc 1252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘𝑢)) = ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩)
152	136, 151	eqtrd 2242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) = ⟨((1st ‘𝑢) + 1), ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢))⟩)
153	152, 144	eqeltrd 2286	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
154	153	ralrimiva 2583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
155	154	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
156		frecsuc 6523	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ ⟨𝑀, (𝐹‘𝑀)⟩ ∈ ((ℤ≥‘𝑀) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
157	155, 121, 81, 156	syl3anc 1252	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
158	133, 157	eqtrid 2254	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)))
159	14	fveq1i 5604	. . . . . . . . . . . . . . 15 ⊢ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)
160	159	fveq2i 5606	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘))
161	158, 160	eqtr4di 2260	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅‘𝑘)))
162	128	fveq2d 5607	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩))
163	161, 162	eqtrd 2242	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩))
164		df-ov 5977	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅‘𝑘))) = ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩)
165	163, 164	eqtr4di 2260	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅‘𝑘))))
166		oveq1 5981	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦))
167	112, 166	opeq12d 3844	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)
168		oveq2 5982	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) = ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘))))
169	168	opeq2d 3843	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩)
170	167, 169, 149	ovmpog 6110	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝑅‘𝑘)) ∈ (ℤ≥‘𝑀) ∧ (2nd ‘(𝑅‘𝑘)) ∈ V ∧ ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅‘𝑘))) = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩)
171	84, 87, 110, 170	syl3anc 1252	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → ((1st ‘(𝑅‘𝑘))(𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩)(2nd ‘(𝑅‘𝑘))) = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩)
172	165, 171	eqtrd 2242	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((1st ‘(𝑅‘𝑘))(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘(𝑅‘𝑘)))⟩)
173	172, 107	eqtrd 2242	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((1st ‘(𝑅‘𝑘)) + 1), ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
174	119, 132, 173	3eqtr4rd 2253	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘))
175	174	exp31 364	. . . . . . 7 ⊢ (𝑘 ∈ ω → (𝜑 → ((𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘) → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘))))
176	175	a2d 26	. . . . . 6 ⊢ (𝑘 ∈ ω → ((𝜑 → (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑘)) → (𝜑 → (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘suc 𝑘))))
177	60, 64, 68, 72, 79, 176	finds 4669	. . . . 5 ⊢ (𝑛 ∈ ω → (𝜑 → (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑛)))
178	177	impcom 125	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → (𝑅‘𝑛) = (frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)‘𝑛))
179	17, 56, 178	eqfnfvd 5708	. . 3 ⊢ (𝜑 → 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
180	179	rneqd 4929	. 2 ⊢ (𝜑 → ran 𝑅 = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
181	1, 180	eqtr4id 2261	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ∀wral 2488  Vcvv 2779   ⊆ wss 3177  ∅c0 3471  ⟨cop 3649  suc csuc 4433  ωcom 4659   × cxp 4694  ran crn 4697   Fn wfn 5289  ⟶wf 5290  ‘cfv 5294  (class class class)co 5974   ∈ cmpo 5976  1^st c1st 6254  2^nd c2nd 6255  freccfrec 6506  1c1 7968   + caddc 7970  ℤcz 9414  ℤ_≥cuz 9690  seqcseq 10636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-inn 9079  df-n0 9338  df-z 9415  df-uz 9691  df-seqfrec 10637

This theorem is referenced by:  seq3-1  10651  seqf  10653  seq3p1  10654