Theorem frecuzrdgg 10351

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 5486	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑅‘𝑤) = (𝑅‘∅))
3	2	fveq2d 5490	. . . . 5 ⊢ (𝑤 = ∅ → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘∅)))
4		fveq2 5486	. . . . 5 ⊢ (𝑤 = ∅ → (𝐺‘𝑤) = (𝐺‘∅))
5	3, 4	eqeq12d 2180	. . . 4 ⊢ (𝑤 = ∅ → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘∅)) = (𝐺‘∅)))
6	5	imbi2d 229	. . 3 ⊢ (𝑤 = ∅ → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))))
7		fveq2 5486	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝑅‘𝑤) = (𝑅‘𝑘))
8	7	fveq2d 5490	. . . . 5 ⊢ (𝑤 = 𝑘 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘𝑘)))
9		fveq2 5486	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝐺‘𝑤) = (𝐺‘𝑘))
10	8, 9	eqeq12d 2180	. . . 4 ⊢ (𝑤 = 𝑘 → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)))
11	10	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘))))
12		fveq2 5486	. . . . . 6 ⊢ (𝑤 = suc 𝑘 → (𝑅‘𝑤) = (𝑅‘suc 𝑘))
13	12	fveq2d 5490	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘suc 𝑘)))
14		fveq2 5486	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (𝐺‘𝑤) = (𝐺‘suc 𝑘))
15	13, 14	eqeq12d 2180	. . . 4 ⊢ (𝑤 = suc 𝑘 → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘)))
16	15	imbi2d 229	. . 3 ⊢ (𝑤 = suc 𝑘 → ((𝜑 → (1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤)) ↔ (𝜑 → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))))
17		fveq2 5486	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑅‘𝑤) = (𝑅‘𝑁))
18	17	fveq2d 5490	. . . . 5 ⊢ (𝑤 = 𝑁 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘𝑁)))
19		fveq2 5486	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐺‘𝑤) = (𝐺‘𝑁))
20	18, 19	eqeq12d 2180	. . . 4 ⊢ (𝑤 = 𝑁 → ((1st ‘(𝑅‘𝑤)) = (𝐺‘𝑤) ↔ (1st ‘(𝑅‘𝑁)) = (𝐺‘𝑁)))
21	20	imbi2d 229	. . 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 6118	. . . . 5 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (1st ‘⟨𝐶, 𝐴⟩) = 𝐶)
25	22, 23, 24	syl2anc 409	. . . 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 5487	. . . . . 6 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
28		opexg 4206	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
29		frec0g 6365	. . . . . . . 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 409	. . . . . 6 ⊢ (𝜑 → (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅) = ⟨𝐶, 𝐴⟩)
32	27, 31	syl5eq 2211	. . . . 5 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
33	32	fveq2d 5490	. . . 4 ⊢ (𝜑 → (1st ‘(𝑅‘∅)) = (1st ‘⟨𝐶, 𝐴⟩))
34		frecuzrdgg.g	. . . . 5 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
35	22, 34	frec2uz0d 10334	. . . 4 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
36	25, 33, 35	3eqtr4d 2208	. . 3 ⊢ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))
37	22, 34	frec2uzf1od 10341	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
38		f1of 5432	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝐺:ω⟶(ℤ≥‘𝐶))
41		simpll 519	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝑘 ∈ ω)
42	40, 41	ffvelrnd 5621	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝐺‘𝑘) ∈ (ℤ≥‘𝐶))
43		peano2uz 9521	. . . . . . . 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 5850	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ((𝐺‘𝑘)𝐹𝑦) = ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))))
46	45	eleq1d 2235	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → (((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆 ↔ ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆))
47		oveq1 5849	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐹𝑦) = ((𝐺‘𝑘)𝐹𝑦))
48	47	eleq1d 2235	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆))
49	48	ralbidv 2466	. . . . . . . . 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 2548	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
52	51	ad2antlr 481	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ∀𝑥 ∈ (ℤ≥‘𝐶)∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆)
53	49, 52, 42	rspcdva 2835	. . . . . . . 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 10350	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
56	55	ad2antlr 481	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝑅:ω⟶((ℤ≥‘𝐶) × 𝑆))
57	56, 41	ffvelrnd 5621	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘𝑘) ∈ ((ℤ≥‘𝐶) × 𝑆))
58		xp2nd 6134	. . . . . . . . 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 2835	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆)
61		op1stg 6118	. . . . . . 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 409	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩) = ((𝐺‘𝑘) + 1))
63		1st2nd2 6143	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
64	63	adantl 275	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
65	64	fveq2d 5490	. . . . . . . . . . . . . 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 5845	. . . . . . . . . . . . . . . 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 6133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
68	67	adantl 275	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (1st ‘𝑧) ∈ (ℤ≥‘𝐶))
69	54	ad3antlr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → 𝑆 ⊆ 𝑇)
70		xp2nd 6134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
71	70	adantl 275	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑆)
72	69, 71	sseldd 3143	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
73		peano2uz 9521	. . . . . . . . . . . . . . . . . . 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 5850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
76	75	eleq1d 2235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
77		oveq1 5849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
78	77	eleq1d 2235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
79	78	ralbidv 2466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (1st ‘𝑧) → (∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
80	51	ad3antlr 485	. . . . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
82	76, 81, 71	rspcdva 2835	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
83		opelxpi 4636	. . . . . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . . . . . 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 5849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
86	85, 77	opeq12d 3766	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
87	75	opeq2d 3765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
88		eqid 2165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
89	86, 87, 88	ovmpog 5976	. . . . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . . 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 2213	. . . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . 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 2539	. . . . . . . . . . . 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 9480	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
96	22, 95	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
97		opelxpi 4636	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
98	96, 23, 97	syl2anc 409	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
99	98	ad2antlr 481	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨𝐶, 𝐴⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
100		frecsuc 6375	. . . . . . . . . . . 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 1228	. . . . . . . . . . 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 5487	. . . . . . . . . . 11 ⊢ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘)
103	26	fveq1i 5487	. . . . . . . . . . . 12 ⊢ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)
104	103	fveq2i 5489	. . . . . . . . . . 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 2224	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑘)))
106		1st2nd2 6143	. . . . . . . . . . . 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 5490	. . . . . . . . . 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 2198	. . . . . . . . 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 109	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘))
111	110	opeq1d 3764	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩ = ⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩)
112	111	fveq2d 5490	. . . . . . . . 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 2198	. . . . . . . 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 5845	. . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝑆 ⊆ 𝑇)
116	115, 59	sseldd 3143	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑇)
117		opelxpi 4636	. . . . . . . . . . 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 409	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩ ∈ ((ℤ≥‘𝐶) × 𝑆))
119		oveq1 5849	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥 + 1) = ((𝐺‘𝑘) + 1))
120	119, 47	opeq12d 3766	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹𝑦)⟩)
121	45	opeq2d 3765	. . . . . . . . . . 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 5976	. . . . . . . . . 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 1228	. . . . . . . . 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 2213	. . . . . . . 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 2198	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
126	125	fveq2d 5490	. . . . . 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 481	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → 𝐶 ∈ ℤ)
128	127, 34, 41	frec2uzsucd 10336	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝐺‘suc 𝑘) = ((𝐺‘𝑘) + 1))
129	62, 126, 128	3eqtr4d 2208	. . . . 5 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (1st ‘(𝑅‘suc 𝑘)) = (𝐺‘suc 𝑘))
130	129	exp31 362	. . . 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 4577	. 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 103   = wceq 1343   ∈ wcel 2136  ∀wral 2444  Vcvv 2726   ⊆ wss 3116  ∅c0 3409  ⟨cop 3579   ↦ cmpt 4043  suc csuc 4343  ωcom 4567   × cxp 4602  ⟶wf 5184  –1-1-onto→wf1o 5187  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  1^st c1st 6106  2^nd c2nd 6107  freccfrec 6358  1c1 7754   + caddc 7756  ℤcz 9191  ℤ_≥cuz 9466

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467

This theorem is referenced by:  frecuzrdgdomlem  10352  frecuzrdgfunlem  10354  frecuzrdgsuctlem  10358