Theorem frecuzrdgg 10508

Description: Lemma for other theorems involving the the recursive definition generator on upper integers. Evaluating 𝑅 at a natural number gives an ordered pair whose first element is the mapping of that natural number via 𝐺. (Contributed by Jim Kingdon, 23-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⟩)
frecuzrdgg.n	⊢ (𝜑 → 𝑁 ∈ ω)
frecuzrdgg.g	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)

Assertion

Ref	Expression
frecuzrdgg	⊢ (𝜑 → (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁))

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

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

Proof of Theorem frecuzrdgg

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

Step	Hyp	Ref	Expression
1		frecuzrdgg.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ω)
2		fveq2 5558	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑅‘𝑤) = (𝑅‘∅))
3	2	fveq2d 5562	. . . . 5 ⊢ (𝑤 = ∅ → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘∅)))
4		fveq2 5558	. . . . 5 ⊢ (𝑤 = ∅ → (𝐺‘𝑤) = (𝐺‘∅))
5	3, 4	eqeq12d 2211	. . . 4 ⊢ (𝑤 = ∅ → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘∅)) = (𝐺‘∅)))
6	5	imbi2d 230	. . 3 ⊢ (𝑤 = ∅ → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))))
7		fveq2 5558	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝑅‘𝑤) = (𝑅‘𝑘))
8	7	fveq2d 5562	. . . . 5 ⊢ (𝑤 = 𝑘 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘𝑘)))
9		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝐺‘𝑤) = (𝐺‘𝑘))
10	8, 9	eqeq12d 2211	. . . 4 ⊢ (𝑤 = 𝑘 → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)))
11	10	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘))))
12		fveq2 5558	. . . . . 6 ⊢ (𝑤 = suc 𝑘 → (𝑅‘𝑤) = (𝑅‘suc 𝑘))
13	12	fveq2d 5562	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘suc 𝑘)))
14		fveq2 5558	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (𝐺‘𝑤) = (𝐺‘suc 𝑘))
15	13, 14	eqeq12d 2211	. . . 4 ⊢ (𝑤 = suc 𝑘 → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘)))
16	15	imbi2d 230	. . 3 ⊢ (𝑤 = suc 𝑘 → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
17		fveq2 5558	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑅‘𝑤) = (𝑅‘𝑁))
18	17	fveq2d 5562	. . . . 5 ⊢ (𝑤 = 𝑁 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘𝑁)))
19		fveq2 5558	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐺‘𝑤) = (𝐺‘𝑁))
20	18, 19	eqeq12d 2211	. . . 4 ⊢ (𝑤 = 𝑁 → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁)))
21	20	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁))))
22		frecuzrdgrclt.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ)
23		frecuzrdgrclt.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
24		op1stg 6208	. . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (1st ‘⟨𝐶, 𝐴⟩) = 𝐶)
25	22, 23, 24	syl2anc 411	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐶, 𝐴⟩) = 𝐶)
26		frecuzrdgrclt.r	. . . . . . 7 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
27	26	fveq1i 5559	. . . . . 6 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
28		opexg 4261	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
29		frec0g 6455	. . . . . . . 8 ⊢ (⟨𝐶, 𝐴⟩ ∈ V → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
30	28, 29	syl 14	. . . . . . 7 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
31	22, 23, 30	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
32	27, 31	eqtrid 2241	. . . . 5 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
33	32	fveq2d 5562	. . . 4 ⊢ (𝜑 → (1st ‘(𝑅‘∅)) = (1st ‘⟨𝐶, 𝐴⟩))
34		frecuzrdgg.g	. . . . 5 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
35	22, 34	frec2uz0d 10491	. . . 4 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
36	25, 33, 35	3eqtr4d 2239	. . 3 ⊢ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))
37	22, 34	frec2uzf1od 10498	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
38		f1of 5504	. . . . . . . . . . 11 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘𝐶) → 𝐺:ω⟶(ℤ≥‘𝐶))
39	37, 38	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ω⟶(ℤ≥‘𝐶))
40	39	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝐺:ω⟶(ℤ≥‘𝐶))
41		simpll 527	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝑘 ∈ ω)
42	40, 41	ffvelcdmd 5698	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝐺‘𝑘) ∈ (ℤ≥‘𝐶))
43		peano2uz 9657	. . . . . . . 8 ⊢ ((𝐺‘𝑘) ∈ (ℤ≥‘𝐶) → ((𝐺‘𝑘) + 1) ∈ (ℤ≥‘𝐶))
44	42, 43	syl 14	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝐺‘𝑘) + 1) ∈ (ℤ≥‘𝐶))
45		oveq2 5930	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ((𝐺‘𝑘)𝐹𝑦) = ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))))
46	45	eleq1d 2265	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → (((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆 ↔ ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆))
47		oveq1 5929	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐹𝑦) = ((𝐺‘𝑘)𝐹𝑦))
48	47	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆))
49	48	ralbidv 2497	. . . . . . . . 9 ⊢ (𝑥 = (𝐺‘𝑘) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆))
50		frecuzrdgrclt.f	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
51	50	ralrimivva 2579	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
52	51	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
53	49, 52, 42	rspcdva 2873	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ∀𝑦 ∈ 𝑆 ((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆)
54		frecuzrdgrclt.t	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ⊆ 𝑇)
55	22, 23, 54, 50, 26	frecuzrdgrclt 10507	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
56	55	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
57	56, 41	ffvelcdmd 5698	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘𝑘) ∈ ((ℤ≥‘𝐶) × 𝑆))
58		xp2nd 6224	. . . . . . . . 9 ⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑆)
59	57, 58	syl 14	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑆)
60	46, 53, 59	rspcdva 2873	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆)
61		op1stg 6208	. . . . . . 7 ⊢ ((((𝐺‘𝑘) + 1) ∈ (ℤ≥‘𝐶) ∧ ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆) → (1st ‘⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩) = ((𝐺‘𝑘) + 1))
62	44, 60, 61	syl2anc 411	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩) = ((𝐺‘𝑘) + 1))
63		1st2nd2 6233	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
64	63	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
65	64	fveq2d 5562	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 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)⟩))
66		df-ov 5925	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
67		xp1st 6223	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
68	67	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
69	54	ad3antlr 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑆 ⊆ 𝑇)
70		xp2nd 6224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
71	70	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
72	69, 71	sseldd 3184	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
73		peano2uz 9657	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑧) ∈ (ℤ≥‘𝐶) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
74	68, 73	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶))
75		oveq2 5930	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
76	75	eleq1d 2265	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
77		oveq1 5929	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
78	77	eleq1d 2265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
79	78	ralbidv 2497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (1st ‘𝑧) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
80	51	ad3antlr 493	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
81	79, 80, 68	rspcdva 2873	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
82	76, 81, 71	rspcdva 2873	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
83		opelxpi 4695	. . . . . . . . . . . . . . . . . 18 ⊢ ((((1st ‘𝑧) + 1) ∈ (ℤ≥‘𝐶) ∧ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
84	74, 82, 83	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
85		oveq1 5929	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
86	85, 77	opeq12d 3816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
87	75	opeq2d 3815	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
88		eqid 2196	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
89	86, 87, 88	ovmpog 6057	. . . . . . . . . . . . . . . . 17 ⊢ (((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))⟩)
90	68, 72, 84, 89	syl3anc 1249	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘𝑧)) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
91	66, 90	eqtr3id 2243	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
92	91, 84	eqeltrd 2273	. . . . . . . . . . . . . 14 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩) ∈ ((ℤ≥‘𝐶) × 𝑆))
93	65, 92	eqeltrd 2273	. . . . . . . . . . . . 13 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
94	93	ralrimiva 2570	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆))
95		uzid 9615	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
96	22, 95	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
97		opelxpi 4695	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
98	96, 23, 97	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
99	98	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
100		frecsuc 6465	. . . . . . . . . . . 12 ⊢ ((∀𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘𝑧) ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆) ∧ 𝑘 ∈ ω) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)))
101	94, 99, 41, 100	syl3anc 1249	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)))
102	26	fveq1i 5559	. . . . . . . . . . 11 ⊢ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘)
103	26	fveq1i 5559	. . . . . . . . . . . 12 ⊢ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)
104	103	fveq2i 5561	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘))
105	101, 102, 104	3eqtr4g 2254	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑘)))
106		1st2nd2 6233	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝑘) ∈ ((ℤ≥‘𝐶) × 𝑆) → (𝑅‘𝑘) = ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩)
107	57, 106	syl 14	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘𝑘) = ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩)
108	107	fveq2d 5562	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑘)) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩))
109	105, 108	eqtrd 2229	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩))
110		simpr 110	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘))
111	110	opeq1d 3814	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩ = ⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩)
112	111	fveq2d 5562	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩))
113	109, 112	eqtrd 2229	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩))
114		df-ov 5925	. . . . . . . . 9 ⊢ ((𝐺‘𝑘)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑘))) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩)
115	54	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝑆 ⊆ 𝑇)
116	115, 59	sseldd 3184	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑇)
117		opelxpi 4695	. . . . . . . . . . 11 ⊢ ((((𝐺‘𝑘) + 1) ∈ (ℤ≥‘𝐶) ∧ ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆) → ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
118	44, 60, 117	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
119		oveq1 5929	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥 + 1) = ((𝐺‘𝑘) + 1))
120	119, 47	opeq12d 3816	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹𝑦)⟩)
121	45	opeq2d 3815	. . . . . . . . . . 11 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹𝑦)⟩ = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
122	120, 121, 88	ovmpog 6057	. . . . . . . . . 10 ⊢ (((𝐺‘𝑘) ∈ (ℤ≥‘𝐶) ∧ (2nd ‘(𝑅‘𝑘)) ∈ 𝑇 ∧ ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((𝐺‘𝑘)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑘))) = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
123	42, 116, 118, 122	syl3anc 1249	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝐺‘𝑘)(𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)(2nd ‘(𝑅‘𝑘))) = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
124	114, 123	eqtr3id 2243	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩) = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
125	113, 124	eqtrd 2229	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
126	125	fveq2d 5562	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘(𝑅‘suc 𝑘)) = (1st ‘⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩))
127	22	ad2antlr 489	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝐶 ∈ ℤ)
128	127, 34, 41	frec2uzsucd 10493	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝐺‘suc 𝑘) = ((𝐺‘𝑘) + 1))
129	62, 126, 128	3eqtr4d 2239	. . . . 5 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))
130	129	exp31 364	. . . 4 ⊢ (𝑘 ∈ ω → (𝜑 → ((1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘) → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
131	130	a2d 26	. . 3 ⊢ (𝑘 ∈ ω → ((𝜑 → (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝜑 → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
132	6, 11, 16, 21, 36, 131	finds 4636	. 2 ⊢ (𝑁 ∈ ω → (𝜑 → (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁)))
133	1, 132	mpcom 36	1 ⊢ (𝜑 → (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ⊆ wss 3157  ∅c0 3450  ⟨cop 3625   ↦ cmpt 4094  suc csuc 4400  ωcom 4626   × cxp 4661  ⟶wf 5254  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  1^st c1st 6196  2^nd c2nd 6197  freccfrec 6448  1c1 7880   + caddc 7882  ℤcz 9326  ℤ_≥cuz 9601

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602

This theorem is referenced by:  frecuzrdgdomlem  10509  frecuzrdgfunlem  10511  frecuzrdgsuctlem  10515