Theorem frecuzrdgsuctlem 10196

Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10172 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 10194	. . . . 5 ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
8	7	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
9		ffun 5275	. . . 4 ⊢ (𝑃:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑃)
10	8, 9	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → Fun 𝑃)
11		1st2nd2 6073	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
12	11	adantl 275	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
13	12	fveq2d 5425	. . . . . . . . . . . . 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 5777	. . . . . . . . . . . . 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	syl6eqr 2190	. . . . . . . . . . . 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 6063	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
17	16	adantl 275	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
18	3	ad2antrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑆 ⊆ 𝑇)
19		xp2nd 6064	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
20	19	adantl 275	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
21	18, 20	sseldd 3098	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
22		peano2uz 9378	. . . . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
25	24	eleq1d 2208	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
26		oveq1 5781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
27	26	eleq1d 2208	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
28	27	ralbidv 2437	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑧) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
29	4	ralrimivva 2514	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
30	29	ad2antrr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
31	28, 30, 17	rspcdva 2794	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
32	25, 31, 20	rspcdva 2794	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
33		opelxpi 4571	. . . . . . . . . . . . . 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 408	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
35		oveq1 5781	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
36	35, 26	opeq12d 3713	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
37	24	opeq2d 3712	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
38		eqid 2139	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
39	36, 37, 38	ovmpog 5905	. . . . . . . . . . . . 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 1216	. . . . . . . . . . . 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 2172	. . . . . . . . . . 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 2216	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
43	42	ralrimiva 2505	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
44		uzid 9340	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
45	1, 44	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
46		opelxpi 4571	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
47	45, 2, 46	syl2anc 408	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
48	47	adantr 274	. . . . . . . . 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 10179	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
51		f1ocnvdm 5682	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
52	50, 51	sylan 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω)
53		frecsuc 6304	. . . . . . . . 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 1216	. . . . . . . 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 5422	. . . . . . . 8 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵))
56	5	fveq1i 5422	. . . . . . . . 9 ⊢ (𝑅‘(◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))
57	56	fveq2i 5424	. . . . . . . 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 2197	. . . . . . 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 10188	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
60	59	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
61	60, 52	ffvelrnd 5556	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆))
62		1st2nd2 6073	. . . . . . . . . 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 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐶 ∈ ℤ)
65	2	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐴 ∈ 𝑆)
66	3	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑆 ⊆ 𝑇)
67	4	adantlr 468	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
68	64, 65, 66, 67, 5, 52, 49	frecuzrdgg 10189	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝐵))) = (𝐺‘(◡𝐺‘𝐵)))
69		f1ocnvfv2 5679	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
70	50, 69	sylan 281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵)
71	68, 70	eqtrd 2172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝐵))) = 𝐵)
72	71	opeq1d 3711	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝐵))), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
73	63, 72	eqtrd 2172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
74	73	fveq2d 5425	. . . . . . 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 2172	. . . . . 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 5777	. . . . . 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	syl6eqr 2190	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = (𝐵(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
78		simpr 109	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝐵 ∈ (ℤ≥‘𝐶))
79		xp2nd 6064	. . . . . . . 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 3098	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑇)
82		peano2uz 9378	. . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
83	82	adantl 275	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵 + 1) ∈ (ℤ≥‘𝐶))
84	67, 78, 80	caovcld 5924	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆)
85		opelxp 4569	. . . . . . 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 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
87		oveq1 5781	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 + 1) = (𝐵 + 1))
88		oveq1 5781	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝐹𝑦) = (𝐵𝐹𝑦))
89	87, 88	opeq12d 3713	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝐵 + 1), (𝐵𝐹𝑦)⟩)
90		oveq2 5782	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → (𝐵𝐹𝑦) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
91	90	opeq2d 3712	. . . . . . 7 ⊢ (𝑦 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ⟨(𝐵 + 1), (𝐵𝐹𝑦)⟩ = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
92	89, 91, 38	ovmpog 5905	. . . . . 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 1216	. . . . 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 2172	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
95		ffun 5275	. . . . . . 7 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → Fun 𝑅)
96	60, 95	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → Fun 𝑅)
97		peano2 4509	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω)
98	52, 97	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ ω)
99		fdm 5278	. . . . . . . 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 2219	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ dom 𝑅)
102		fvelrn 5551	. . . . . 6 ⊢ ((Fun 𝑅 ∧ suc (◡𝐺‘𝐵) ∈ dom 𝑅) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ ran 𝑅)
103	96, 101, 102	syl2anc 408	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ ran 𝑅)
104	6	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑃 = ran 𝑅)
105	103, 104	eleqtrrd 2219	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ 𝑃)
106	94, 105	eqeltrrd 2217	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ 𝑃)
107		funopfv 5461	. . 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 2219	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ dom 𝑅)
110		fvelrn 5551	. . . . . . 7 ⊢ ((Fun 𝑅 ∧ (◡𝐺‘𝐵) ∈ dom 𝑅) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
111	96, 109, 110	syl2anc 408	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
112	111, 104	eleqtrrd 2219	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ 𝑃)
113	73, 112	eqeltrrd 2217	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑃)
114		funopfv 5461	. . . 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 5790	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵𝐹(𝑃‘𝐵)) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
117	108, 116	eqtr4d 2175	1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ⊆ wss 3071  ⟨cop 3530   ↦ cmpt 3989  suc csuc 4287  ωcom 4504   × cxp 4537  ^◡ccnv 4538  dom cdm 4539  ran crn 4540  Fun wfun 5117  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776  1^st c1st 6036  2^nd c2nd 6037  freccfrec 6287  1c1 7621   + caddc 7623  ℤcz 9054  ℤ_≥cuz 9326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327

This theorem is referenced by:  frecuzrdgsuct  10197