Theorem frecuzrdgsuctlem 10300

Description: Successor value of a recursive definition generator on upper integers. See comment in frec2uz0d 10276 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 10298	. . . . 5 ⊢ (𝜑 → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
8	7	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → 𝑃:(ℤ≥‘𝐶)⟶𝑆)
9		ffun 5315	. . . 4 ⊢ (𝑃:(ℤ≥‘𝐶)⟶𝑆 → Fun 𝑃)
10	8, 9	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → Fun 𝑃)
11		1st2nd2 6113	. . . . . . . . . . . . . . 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 5465	. . . . . . . . . . . . 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 5817	. . . . . . . . . . . . 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 2205	. . . . . . . . . . . 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 6103	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑆 ⊆ 𝑇)
19		xp2nd 6104	. . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
22		peano2uz 9473	. . . . . . . . . . . . . . 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 5822	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
25	24	eleq1d 2223	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
26		oveq1 5821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
27	26	eleq1d 2223	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
28	27	ralbidv 2454	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (1st ‘𝑧) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
29	4	ralrimivva 2536	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
30	29	ad2antrr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
31	28, 30, 17	rspcdva 2818	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
32	25, 31, 20	rspcdva 2818	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
33		opelxpi 4611	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
35		oveq1 5821	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
36	35, 26	opeq12d 3745	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
37	24	opeq2d 3744	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
38		eqid 2154	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
39	36, 37, 38	ovmpog 5945	. . . . . . . . . . . . 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 1217	. . . . . . . . . . . 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 2187	. . . . . . . . . . 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 2231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
43	42	ralrimiva 2527	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
44		uzid 9432	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
45	1, 44	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
46		opelxpi 4611	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
47	45, 2, 46	syl2anc 409	. . . . . . . . . 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 10283	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
51		f1ocnvdm 5722	. . . . . . . . . 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 6344	. . . . . . . . 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 1217	. . . . . . . 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 5462	. . . . . . . 8 ⊢ (𝑅‘suc (◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc (◡𝐺‘𝐵))
56	5	fveq1i 5462	. . . . . . . . 9 ⊢ (𝑅‘(◡𝐺‘𝐵)) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘(◡𝐺‘𝐵))
57	56	fveq2i 5464	. . . . . . . 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 2212	. . . . . . 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 10292	. . . . . . . . . . . 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 5596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ((ℤ≥‘𝐶) × 𝑆))
62		1st2nd2 6113	. . . . . . . . . 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 469	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
68	64, 65, 66, 67, 5, 52, 49	frecuzrdgg 10293	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝐵))) = (𝐺‘(◡𝐺‘𝐵)))
69		f1ocnvfv2 5719	. . . . . . . . . . . 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 2187	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (1st ‘(𝑅‘(◡𝐺‘𝐵))) = 𝐵)
72	71	opeq1d 3743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(1st ‘(𝑅‘(◡𝐺‘𝐵))), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
73	63, 72	eqtrd 2187	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) = ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩)
74	73	fveq2d 5465	. . . . . . 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 2187	. . . . . 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 5817	. . . . . 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 2205	. . . . 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 6104	. . . . . . . 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 3125	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ 𝑇)
82		peano2uz 9473	. . . . . . . 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 5964	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ 𝑆)
85		opelxp 4609	. . . . . . 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 414	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
87		oveq1 5821	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 + 1) = (𝐵 + 1))
88		oveq1 5821	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥𝐹𝑦) = (𝐵𝐹𝑦))
89	87, 88	opeq12d 3745	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨(𝐵 + 1), (𝐵𝐹𝑦)⟩)
90		oveq2 5822	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → (𝐵𝐹𝑦) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
91	90	opeq2d 3744	. . . . . . 7 ⊢ (𝑦 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ⟨(𝐵 + 1), (𝐵𝐹𝑦)⟩ = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
92	89, 91, 38	ovmpog 5945	. . . . . 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 1217	. . . . 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 2187	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) = ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩)
95		ffun 5315	. . . . . . 7 ⊢ (𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆) → Fun 𝑅)
96	60, 95	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → Fun 𝑅)
97		peano2 4548	. . . . . . . 8 ⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω)
98	52, 97	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ ω)
99		fdm 5318	. . . . . . . 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 2234	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → suc (◡𝐺‘𝐵) ∈ dom 𝑅)
102		fvelrn 5591	. . . . . 6 ⊢ ((Fun 𝑅 ∧ suc (◡𝐺‘𝐵) ∈ dom 𝑅) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ ran 𝑅)
103	96, 101, 102	syl2anc 409	. . . . 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 2234	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘suc (◡𝐺‘𝐵)) ∈ 𝑃)
106	94, 105	eqeltrrd 2232	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨(𝐵 + 1), (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))⟩ ∈ 𝑃)
107		funopfv 5501	. . 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 2234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ dom 𝑅)
110		fvelrn 5591	. . . . . . 7 ⊢ ((Fun 𝑅 ∧ (◡𝐺‘𝐵) ∈ dom 𝑅) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
111	96, 109, 110	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ ran 𝑅)
112	111, 104	eleqtrrd 2234	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑅‘(◡𝐺‘𝐵)) ∈ 𝑃)
113	73, 112	eqeltrrd 2232	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ 𝑃)
114		funopfv 5501	. . . 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 5830	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐵𝐹(𝑃‘𝐵)) = (𝐵𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))))
117	108, 116	eqtr4d 2190	1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125  ∀wral 2432   ⊆ wss 3098  ⟨cop 3559   ↦ cmpt 4021  suc csuc 4320  ωcom 4543   × cxp 4577  ^◡ccnv 4578  dom cdm 4579  ran crn 4580  Fun wfun 5157  ⟶wf 5159  –1-1-onto→wf1o 5162  ‘cfv 5163  (class class class)co 5814   ∈ cmpo 5816  1^st c1st 6076  2^nd c2nd 6077  freccfrec 6327  1c1 7712   + caddc 7714  ℤcz 9146  ℤ_≥cuz 9418

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-inn 8813  df-n0 9070  df-z 9147  df-uz 9419

This theorem is referenced by:  frecuzrdgsuct  10301