Theorem frecuzrdgsuc 10829

Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10814 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 28-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	⊢ (𝜑 → 𝐶 ∈ ℤ)
frec2uz.2	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgrrn.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrrn.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrrn.2	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgtcl.3	⊢ (𝜑 → 𝑇 = ran 𝑅)

Assertion

Ref	Expression
frecuzrdgsuc	⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇‘𝐵)))

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

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

Proof of Theorem frecuzrdgsuc

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

Step	Hyp	Ref	Expression
1		frec2uz.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℤ)
2	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
3		frec2uz.2	. . . . . 6 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
4		frecuzrdgrrn.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
5	4	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
6		frecuzrdgrrn.f	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
7	6	adantlr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
8		frecuzrdgrrn.2	. . . . . 6 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
9		peano2uz 9962	. . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
10	9	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
11	2, 3, 5, 7, 8, 10	frecuzrdglem 10826	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ ran 𝑅)
12		frecuzrdgtcl.3	. . . . . 6 ⊢ (𝜑 → 𝑇 = ran 𝑅)
13	12	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑇 = ran 𝑅)
14	11, 13	eleqtrrd 2318	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇)
15	1, 3, 4, 6, 8, 12	frecuzrdgtcl 10827	. . . . . . 7 ⊢ (𝜑 → 𝑇:(ℤ≥‘𝐶)⟶𝑆)
16		ffun 5531	. . . . . . 7 ⊢ (𝑇:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑇)
17	15, 16	syl 14	. . . . . 6 ⊢ (𝜑 → Fun 𝑇)
18		funopfv 5734	. . . . . 6 ⊢ (Fun 𝑇 → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇 → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))))
19	17, 18	syl 14	. . . . 5 ⊢ (𝜑 → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇 → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))))
20	19	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (⟨(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))⟩ ∈ 𝑇 → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))))
21	14, 20	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))))
22	1, 3	frec2uzf1od 10821	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
23		f1ocnvdm 5977	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
24	22, 23	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
25	2, 3, 24	frec2uzsucd 10816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
26		f1ocnvfv2 5974	. . . . . . . . 9 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
27	22, 26	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
28	27	oveq1d 6090	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝐺‘(◡𝐺‘𝐵)) + 1) = (𝐵 + 1))
29	25, 28	eqtrd 2271	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1))
30		peano2 4737	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω)
31	24, 30	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ ω)
32		f1ocnvfv 5975	. . . . . . 7 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ suc (◡𝐺‘𝐵) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵)))
33	22, 31, 32	syl2an2r 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵)))
34	29, 33	mpd 13	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵))
35	34	fveq2d 5694	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘(𝐵 + 1))) = (𝑅‘suc (◡𝐺‘𝐵)))
36	35	fveq2d 5694	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
37	21, 36	eqtrd 2271	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))))
38		1st2nd2 6399	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
39	38	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
40	39	fveq2d 5694	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
41		df-ov 6078	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
42		xp1st 6389	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
43	42	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
44		xp2nd 6390	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
45	44	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
46		peano2uz 9962	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑧) ∈ (ℤ≥‘𝐶) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
47	43, 46	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
48		oveq2 6083	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
49	48	eleq1d 2307	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
50		oveq1 6082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
51	50	eleq1d 2307	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
52	51	ralbidv 2550	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘𝑧) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
53	6	ralrimivva 2632	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
54	53	ad2antrr 492	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
55	52, 54, 43	rspcdva 2934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
56	49, 55, 45	rspcdva 2934	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
57		opelxp 4799	. . . . . . . . . . . . 13 ⊢ (⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶) ∧ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
58	47, 56, 57	sylanbrc 421	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
59		oveq1 6082	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
60	59, 50	opeq12d 3907	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
61	48	opeq2d 3906	. . . . . . . . . . . . 13 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
62		eqid 2238	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
63	60, 61, 62	ovmpog 6213	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑆 ∧ ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
64	43, 45, 58, 63	syl3anc 1278	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
65	41, 64	eqtr3id 2285	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
66	65, 58	eqeltrd 2315	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) ∈ ((ℤ≥‘𝐶) × 𝑆))
67	40, 66	eqeltrd 2315	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
68	67	ralrimiva 2623	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
69		uzid 9915	. . . . . . . . 9 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
70	2, 69	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ (ℤ≥‘𝐶))
71		opelxp 4799	. . . . . . . 8 ⊢ (⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆))
72	70, 5, 71	sylanbrc 421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
73		frecsuc 6668	. . . . . . 7 ⊢ ((∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ (◡𝐺‘𝐵) ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))))
74	68, 72, 24, 73	syl3anc 1278	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))))
75	8	fveq1i 5691	. . . . . 6 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵))
76	8	fveq1i 5691	. . . . . . 7 ⊢ (𝑅‘(◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))
77	76	fveq2i 5693	. . . . . 6 ⊢ ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵)))
78	74, 75, 77	3eqtr4g 2296	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))))
79	2, 3, 5, 7, 8, 24	frec2uzrdg 10824	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
80	79	fveq2d 5694	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩))
81		df-ov 6078	. . . . . 6 ⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
82	80, 81	eqtr4di 2289	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
83	2, 3, 24	frec2uzuzd 10817	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) ∈ (ℤ≥‘𝐶))
84	2, 3, 5, 7, 8	frecuzrdgrrn 10823	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (◡𝐺‘𝐵) ∈ ω) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆))
85	24, 84	mpdan 425	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆))
86		xp2nd 6390	. . . . . . 7 ⊢ ((𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆)
87	85, 86	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆)
88	28, 10	eqeltrd 2315	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ (ℤ≥‘𝐶))
89	7	caovclg 6232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (𝑧 ∈ (ℤ≥‘𝐶) ∧ 𝑤 ∈ 𝑆)) → (𝑧𝐹𝑤) ∈ 𝑆)
90	89, 83, 87	caovcld 6233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆)
91		opelxp 4799	. . . . . . 7 ⊢ (⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ (ℤ≥‘𝐶) ∧ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆))
92	88, 90, 91	sylanbrc 421	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
93		oveq1 6082	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 + 1) = ((𝐺‘(◡𝐺‘𝐵)) + 1))
94		oveq1 6082	. . . . . . . 8 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤))
95	93, 94	opeq12d 3907	. . . . . . 7 ⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩)
96		oveq2 6083	. . . . . . . 8 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
97	96	opeq2d 3906	. . . . . . 7 ⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)⟩ = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
98		oveq1 6082	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
99		oveq1 6082	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦))
100	98, 99	opeq12d 3907	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩)
101		oveq2 6083	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤))
102	101	opeq2d 3906	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ⟨(𝑧 + 1), (𝑧𝐹𝑦)⟩ = ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
103	100, 102	cbvmpov 6158	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑧 ∈ (ℤ≥‘𝐶), 𝑤 ∈ 𝑆 ↦ ⟨(𝑧 + 1), (𝑧𝐹𝑤)⟩)
104	95, 97, 103	ovmpog 6213	. . . . . 6 ⊢ (((𝐺‘(◡𝐺‘𝐵)) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆 ∧ ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
105	83, 87, 92, 104	syl3anc 1278	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
106	78, 82, 105	3eqtrd 2275	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
107	106	fveq2d 5694	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩))
108		op2ndg 6375	. . . 4 ⊢ ((((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ (ℤ≥‘𝐶) ∧ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆) → (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
109	88, 90, 108	syl2anc 415	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘⟨((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
110	107, 109	eqtrd 2271	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
111		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐵 ∈ (ℤ≥‘𝐶))
112	2, 3, 5, 7, 8, 111	frecuzrdglem 10826	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
113	112, 13	eleqtrrd 2318	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑇)
114		funopfv 5734	. . . . . . 7 ⊢ (Fun 𝑇 → (⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑇 → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
115	17, 114	syl 14	. . . . . 6 ⊢ (𝜑 → (⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑇 → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
116	115	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑇 → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
117	113, 116	mpd 13	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))
118	117	eqcomd 2244	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑇‘𝐵))
119	27, 118	oveq12d 6093	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑇‘𝐵)))
120	37, 110, 119	3eqtrd 2275	1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑇‘(𝐵 + 1)) = (𝐵𝐹(𝑇‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ⟨cop 3708   ↦ cmpt 4187  suc csuc 4505  ωcom 4732   × cxp 4767  ^◡ccnv 4768  ran crn 4770  Fun wfun 5366  ⟶wf 5368  –1-1-onto→wf1o 5371  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  1^st c1st 6362  2^nd c2nd 6363  freccfrec 6651  1c1 8170   + caddc 8172  ℤcz 9623  ℤ_≥cuz 9900

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by: (None)