Theorem seq3val 10460

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 10463, seq3-1 10462 and seq3p1 10464, 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 10448	. 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 5517	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀))
4	3	eleq1d 2246	. . . . . . 7 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆))
5		seq3val.f	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
6	5	ralrimiva 2550	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
7		uzid 9544	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
8	2, 7	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
9	4, 6, 8	rspcdva 2848	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆)
10		ssv 3179	. . . . . . 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 10458	. . . . . 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 10417	. . . . 5 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆))
16		ffn 5367	. . . . 5 ⊢ (𝑅:ω⟶((ℤ≥‘𝑀) × 𝑆) → 𝑅 Fn ω)
17	15, 16	syl 14	. . . 4 ⊢ (𝜑 → 𝑅 Fn ω)
18		1st2nd2 6178	. . . . . . . . . . . 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 5521	. . . . . . . . . 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 5880	. . . . . . . . . 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 2228	. . . . . . . . 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 6168	. . . . . . . . . . 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 6169	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆) → (2nd ‘𝑢) ∈ 𝑆)
26	25	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (2nd ‘𝑢) ∈ 𝑆)
27	26	elexd 2752	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (2nd ‘𝑢) ∈ V)
28		peano2uz 9585	. . . . . . . . . . . 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 6029	. . . . . . . . . . . . 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 5517	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((1st ‘𝑢) + 1) → (𝐹‘𝑥) = (𝐹‘((1st ‘𝑢) + 1)))
33	32	eleq1d 2246	. . . . . . . . . . . . 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 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → (𝐹‘((1st ‘𝑢) + 1)) ∈ 𝑆)
36	31, 26, 35	caovcld 6030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))) ∈ 𝑆)
37		opelxpi 4660	. . . . . . . . . . 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 5884	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑢) → (𝑥 + 1) = ((1st ‘𝑢) + 1))
40		fvoveq1 5900	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑢) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘𝑢) + 1)))
41	40	oveq2d 5893	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘𝑢) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘𝑢) + 1))))
42	39, 41	opeq12d 3788	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘𝑢) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘𝑢) + 1), (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))⟩)
43		oveq1 5884	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘𝑢) → (𝑦 + (𝐹‘((1st ‘𝑢) + 1))) = ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))))
44	43	opeq2d 3787	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘𝑢) → ⟨((1st ‘𝑢) + 1), (𝑦 + (𝐹‘((1st ‘𝑢) + 1)))⟩ = ⟨((1st ‘𝑢) + 1), ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1)))⟩)
45		eqid 2177	. . . . . . . . . . 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 6011	. . . . . . . . . 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 1238	. . . . . . . . 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 2210	. . . . . . . 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 2254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
50	49	ralrimiva 2550	. . . . . 6 ⊢ (𝜑 → ∀𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)‘𝑢) ∈ ((ℤ≥‘𝑀) × 𝑆))
51		opelxpi 4660	. . . . . . 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 6408	. . . . 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 5367	. . . . 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 5517	. . . . . . . 8 ⊢ (𝑐 = ∅ → (𝑅‘𝑐) = (𝑅‘∅))
58		fveq2 5517	. . . . . . . 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 2192	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑐 = 𝑘 → (𝑅‘𝑐) = (𝑅‘𝑘))
62		fveq2 5517	. . . . . . . 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 2192	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑐 = suc 𝑘 → (𝑅‘𝑐) = (𝑅‘suc 𝑘))
66		fveq2 5517	. . . . . . . 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 2192	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑐 = 𝑛 → (𝑅‘𝑐) = (𝑅‘𝑛))
70		fveq2 5517	. . . . . . . 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 2192	. . . . . . 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 5518	. . . . . . . 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 6400	. . . . . . . . 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 2222	. . . . . . 7 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝑀, (𝐹‘𝑀)⟩)
77		frec0g 6400	. . . . . . . 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 2213	. . . . . 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 5654	. . . . . . . . . . 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 6168	. . . . . . . . . . 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 6169	. . . . . . . . . . . 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 2752	. . . . . . . . . 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 5517	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = ((1st ‘(𝑅‘𝑘)) + 1) → (𝐹‘𝑎) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))
91	90	eleq1d 2246	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = ((1st ‘(𝑅‘𝑘)) + 1) → ((𝐹‘𝑎) ∈ 𝑆 ↔ (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)) ∈ 𝑆))
92		fveq2 5517	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
93	92	eleq1d 2246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑎) ∈ 𝑆))
94	93	cbvralv 2705	. . . . . . . . . . . . . . . . 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 9585	. . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . 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 6030	. . . . . . . . . . . . 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 5900	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (1st ‘(𝑅‘𝑘)) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))
102	101	oveq2d 5893	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (1st ‘(𝑅‘𝑘)) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
103		oveq1 5884	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (2nd ‘(𝑅‘𝑘)) → (𝑤 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
104		eqid 2177	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))
105	102, 103, 104	ovmpog 6011	. . . . . . . . . . . . 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 1238	. . . . . . . . . . . 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 3787	. . . . . . . . . . 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 2254	. . . . . . . . . . . 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 4660	. . . . . . . . . . . 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 2255	. . . . . . . . . 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 5884	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑥 + 1) = ((1st ‘(𝑅‘𝑘)) + 1))
113		fvoveq1 5900	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝐹‘(𝑥 + 1)) = (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))
114	113	oveq2d 5893	. . . . . . . . . . . 12 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
115	112, 114	opeq12d 3788	. . . . . . . . . . 11 ⊢ (𝑥 = (1st ‘(𝑅‘𝑘)) → ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩ = ⟨((1st ‘(𝑅‘𝑘)) + 1), (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1)))⟩)
116		oveq1 5884	. . . . . . . . . . . 12 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → (𝑦 + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))) = ((2nd ‘(𝑅‘𝑘)) + (𝐹‘((1st ‘(𝑅‘𝑘)) + 1))))
117	116	opeq2d 3787	. . . . . . . . . . 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 6011	. . . . . . . . . 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 1238	. . . . . . . . 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 6410	. . . . . . . . . . . 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 1238	. . . . . . . . . . 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 5521	. . . . . . . . . . 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 2213	. . . . . . . . . 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 6178	. . . . . . . . . . . . 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 5521	. . . . . . . . . . 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 5880	. . . . . . . . . . 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 2228	. . . . . . . . . 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 2210	. . . . . . . . 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 5518	. . . . . . . . . . . . . . 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 5521	. . . . . . . . . . . . . . . . . . . . 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 5880	. . . . . . . . . . . . . . . . . . . . 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 2228	. . . . . . . . . . . . . . . . . . . 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 5900	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = (1st ‘𝑢) → (𝐹‘(𝑧 + 1)) = (𝐹‘((1st ‘𝑢) + 1)))
138	137	oveq2d 5893	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (1st ‘𝑢) → (𝑤 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘((1st ‘𝑢) + 1))))
139		oveq1 5884	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = (2nd ‘𝑢) → (𝑤 + (𝐹‘((1st ‘𝑢) + 1))) = ((2nd ‘𝑢) + (𝐹‘((1st ‘𝑢) + 1))))
140	138, 139, 104	ovmpog 6011	. . . . . . . . . . . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ ((ℤ≥‘𝑀) × 𝑆)) → ((1st ‘𝑢)(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))(2nd ‘𝑢)) ∈ 𝑆)
143		opelxpi 4660	. . . . . . . . . . . . . . . . . . . . . 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 5884	. . . . . . . . . . . . . . . . . . . . . . 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 3788	. . . . . . . . . . . . . . . . . . . . . 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 5885	. . . . . . . . . . . . . . . . . . . . . . 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 3787	. . . . . . . . . . . . . . . . . . . . . 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 2177	. . . . . . . . . . . . . . . . . . . . . 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 6011	. . . . . . . . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . . . . . . 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 2210	. . . . . . . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . . . . . 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 2550	. . . . . . . . . . . . . . . . 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 6410	. . . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . 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 2222	. . . . . . . . . . . . . 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 5518	. . . . . . . . . . . . . . 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 5520	. . . . . . . . . . . . . 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 2228	. . . . . . . . . . . . 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 5521	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 5880	. . . . . . . . . . . 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 2228	. . . . . . . . . . 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 5884	. . . . . . . . . . . . . 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 3788	. . . . . . . . . . . . 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 5885	. . . . . . . . . . . . . 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 3787	. . . . . . . . . . . . 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 6011	. . . . . . . . . . . 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 1238	. . . . . . . . . . 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 2210	. . . . . . . . . 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 2210	. . . . . . . . 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 2221	. . . . . . . 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 4601	. . . . 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 5618	. . 3 ⊢ (𝜑 → 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
180	179	rneqd 4858	. 2 ⊢ (𝜑 → ran 𝑅 = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩))
181	1, 180	eqtr4id 2229	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2739   ⊆ wss 3131  ∅c0 3424  ⟨cop 3597  suc csuc 4367  ωcom 4591   × cxp 4626  ran crn 4629   Fn wfn 5213  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877   ∈ cmpo 5879  1^st c1st 6141  2^nd c2nd 6142  freccfrec 6393  1c1 7814   + caddc 7816  ℤcz 9255  ℤ_≥cuz 9530  seqcseq 10447

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-seqfrec 10448

This theorem is referenced by:  seq3-1  10462  seqf  10463  seq3p1  10464