Theorem frecuzrdgsuctlem 10684

Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10660 for the description of 𝐺 as the mapping from ω to (ℤ_≥‘𝐶). (Contributed by Jim Kingdon, 29-Apr-2022.)

Hypotheses

Ref	Expression
frecuzrdgrclt.c	⊢ (𝜑 → 𝐶 ∈ ℤ)
frecuzrdgrclt.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
frecuzrdgrclt.t	⊢ (𝜑 → 𝑆 ⊆ 𝑇)
frecuzrdgrclt.f	⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrclt.r	⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgsuctlem.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
frecuzrdgsuctlem.ran	⊢ (𝜑 → 𝑃 = ran 𝑅)

Assertion

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

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

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑃(𝑥,𝑦)

Proof of Theorem frecuzrdgsuctlem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecuzrdgrclt.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ)
2		frecuzrdgrclt.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		frecuzrdgrclt.t	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
4		frecuzrdgrclt.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5		frecuzrdgrclt.r	. . . . . 6 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
6		frecuzrdgsuctlem.ran	. . . . . 6 ⊢ (𝜑 → 𝑃 = ran 𝑅)
7	1, 2, 3, 4, 5, 6	frecuzrdgtclt 10682	. . . . 5 ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
8	7	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
9		ffun 5485	. . . 4 ⊢ (𝑃:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑃)
10	8, 9	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → Fun 𝑃)
11		1st2nd2 6337	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
12	11	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
13	12	fveq2d 5643	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
14		df-ov 6020	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
15	13, 14	eqtr4di 2282	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)))
16		xp1st 6327	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
17	16	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
18	3	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑆 ⊆ 𝑇)
19		xp2nd 6328	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
20	19	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
21	18, 20	sseldd 3228	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
22		peano2uz 9816	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑧) ∈ (ℤ≥‘𝐶) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
23	17, 22	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
24		oveq2 6025	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
25	24	eleq1d 2300	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
26		oveq1 6024	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
27	26	eleq1d 2300	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
28	27	ralbidv 2532	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑧) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
29	4	ralrimivva 2614	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
30	29	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
31	28, 30, 17	rspcdva 2915	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
32	25, 31, 20	rspcdva 2915	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
33		opelxpi 4757	. . . . . . . . . . . . . 14 ⊢ ((((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶) ∧ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
34	23, 32, 33	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
35		oveq1 6024	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
36	35, 26	opeq12d 3870	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
37	24	opeq2d 3869	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
38		eqid 2231	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
39	36, 37, 38	ovmpog 6155	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑧) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘𝑧) ∈ 𝑇 ∧ ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
40	17, 21, 34, 39	syl3anc 1273	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
41	15, 40	eqtrd 2264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
42	41, 34	eqeltrd 2308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
43	42	ralrimiva 2605	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
44		uzid 9769	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
45	1, 44	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
46		opelxpi 4757	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
47	45, 2, 46	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
48	47	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
49		frecuzrdgsuctlem.g	. . . . . . . . . . 11 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
50	1, 49	frec2uzf1od 10667	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
51		f1ocnvdm 5921	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
52	50, 51	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
53		frecsuc 6572	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ (◡𝐺‘𝐵) ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))))
54	43, 48, 52, 53	syl3anc 1273	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))))
55	5	fveq1i 5640	. . . . . . . 8 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵))
56	5	fveq1i 5640	. . . . . . . . 9 ⊢ (𝑅‘(◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))
57	56	fveq2i 5642	. . . . . . . 8 ⊢ ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵)))
58	54, 55, 57	3eqtr4g 2289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))))
59	1, 2, 3, 4, 5	frecuzrdgrclt 10676	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
60	59	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
61	60, 52	ffvelcdmd 5783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆))
62		1st2nd2 6337	. . . . . . . . . 10 ⊢ ((𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(1st ‘(𝑅‘(◡𝐺‘𝐵))), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
63	61, 62	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) = ⟨(1st ‘(𝑅‘(◡𝐺‘𝐵))), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
64	1	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
65	2	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
66	3	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑆 ⊆ 𝑇)
67	4	adantlr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
68	64, 65, 66, 67, 5, 52, 49	frecuzrdgg 10677	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝐵))) = (𝐺‘(◡𝐺‘𝐵)))
69		f1ocnvfv2 5918	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
70	50, 69	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
71	68, 70	eqtrd 2264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝐵))) = 𝐵)
72	71	opeq1d 3868	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝐵))), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
73	63, 72	eqtrd 2264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
74	73	fveq2d 5643	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩))
75	58, 74	eqtrd 2264	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩))
76		df-ov 6020	. . . . . 6 ⊢ (𝐵(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
77	75, 76	eqtr4di 2282	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = (𝐵(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
78		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐵 ∈ (ℤ≥‘𝐶))
79		xp2nd 6328	. . . . . . . 8 ⊢ ((𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆)
80	61, 79	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑆)
81	66, 80	sseldd 3228	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑇)
82		peano2uz 9816	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
83	82	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
84	67, 78, 80	caovcld 6175	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆)
85		opelxp 4755	. . . . . . 7 ⊢ (⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ ((𝐵 + 1) ∈ (ℤ≥‘𝐶) ∧ (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆))
86	83, 84, 85	sylanbrc 417	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
87		oveq1 6024	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 + 1) = (𝐵 + 1))
88		oveq1 6024	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝐹𝑦) = (𝐵𝐹𝑦))
89	87, 88	opeq12d 3870	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝐵 + 1), (𝐵𝐹𝑦)⟩)
90		oveq2 6025	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → (𝐵𝐹𝑦) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
91	90	opeq2d 3869	. . . . . . 7 ⊢ (𝑦 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ⟨(𝐵 + 1), (𝐵𝐹𝑦)⟩ = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
92	89, 91, 38	ovmpog 6155	. . . . . 6 ⊢ ((𝐵 ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑇 ∧ ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → (𝐵(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
93	78, 81, 86, 92	syl3anc 1273	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
94	77, 93	eqtrd 2264	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
95		ffun 5485	. . . . . . 7 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → Fun 𝑅)
96	60, 95	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → Fun 𝑅)
97		peano2 4693	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω)
98	52, 97	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ ω)
99		fdm 5488	. . . . . . . 8 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → dom 𝑅 = ω)
100	60, 99	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → dom 𝑅 = ω)
101	98, 100	eleqtrrd 2311	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ dom 𝑅)
102		fvelrn 5778	. . . . . 6 ⊢ ((Fun 𝑅 ∧ suc (◡𝐺‘𝐵) ∈ dom 𝑅) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ ran 𝑅)
103	96, 101, 102	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ ran 𝑅)
104	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑃 = ran 𝑅)
105	103, 104	eleqtrrd 2311	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ 𝑃)
106	94, 105	eqeltrrd 2309	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ 𝑃)
107		funopfv 5683	. . 3 ⊢ (Fun 𝑃 → (⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ 𝑃 → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))))
108	10, 106, 107	sylc 62	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
109	52, 100	eleqtrrd 2311	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ dom 𝑅)
110		fvelrn 5778	. . . . . . 7 ⊢ ((Fun 𝑅 ∧ (◡𝐺‘𝐵) ∈ dom 𝑅) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
111	96, 109, 110	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
112	111, 104	eleqtrrd 2311	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ 𝑃)
113	73, 112	eqeltrrd 2309	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑃)
114		funopfv 5683	. . . 4 ⊢ (Fun 𝑃 → (⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑃 → (𝑃‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
115	10, 113, 114	sylc 62	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))
116	115	oveq2d 6033	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵𝐹(𝑃‘𝐵)) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
117	108, 116	eqtr4d 2267	1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  ⟨cop 3672   ↦ cmpt 4150  suc csuc 4462  ωcom 4688   × cxp 4723  ^◡ccnv 4724  dom cdm 4725  ran crn 4726  Fun wfun 5320  ⟶wf 5322  –1-1-onto→wf1o 5325  ‘cfv 5326  (class class class)co 6017   ∈ cmpo 6019  1^st c1st 6300  2^nd c2nd 6301  freccfrec 6555  1c1 8032   + caddc 8034  ℤcz 9478  ℤ_≥cuz 9754

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755

This theorem is referenced by:  frecuzrdgsuct  10685