Theorem frecuzrdgg 10487

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 5554	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑅‘𝑤) = (𝑅‘∅))
3	2	fveq2d 5558	. . . . 5 ⊢ (𝑤 = ∅ → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘∅)))
4		fveq2 5554	. . . . 5 ⊢ (𝑤 = ∅ → (𝐺‘𝑤) = (𝐺‘∅))
5	3, 4	eqeq12d 2208	. . . 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 5554	. . . . . 6 ⊢ (𝑤 = 𝑘 → (𝑅‘𝑤) = (𝑅‘𝑘))
8	7	fveq2d 5558	. . . . 5 ⊢ (𝑤 = 𝑘 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘𝑘)))
9		fveq2 5554	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝐺‘𝑤) = (𝐺‘𝑘))
10	8, 9	eqeq12d 2208	. . . 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 5554	. . . . . 6 ⊢ (𝑤 = suc 𝑘 → (𝑅‘𝑤) = (𝑅‘suc 𝑘))
13	12	fveq2d 5558	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘suc 𝑘)))
14		fveq2 5554	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (𝐺‘𝑤) = (𝐺‘suc 𝑘))
15	13, 14	eqeq12d 2208	. . . 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 5554	. . . . . 6 ⊢ (𝑤 = 𝑁 → (𝑅‘𝑤) = (𝑅‘𝑁))
18	17	fveq2d 5558	. . . . 5 ⊢ (𝑤 = 𝑁 → (1st ‘(𝑅‘𝑤)) = (1st ‘(𝑅‘𝑁)))
19		fveq2 5554	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝐺‘𝑤) = (𝐺‘𝑁))
20	18, 19	eqeq12d 2208	. . . 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 6203	. . . . 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 5555	. . . . . 6 ⊢ (𝑅‘∅) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘∅)
28		opexg 4257	. . . . . . . 8 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → ⟨𝐶, 𝐴⟩ ∈ V)
29		frec0g 6450	. . . . . . . 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 2238	. . . . 5 ⊢ (𝜑 → (𝑅‘∅) = ⟨𝐶, 𝐴⟩)
33	32	fveq2d 5558	. . . 4 ⊢ (𝜑 → (1st ‘(𝑅‘∅)) = (1st ‘⟨𝐶, 𝐴⟩))
34		frecuzrdgg.g	. . . . 5 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
35	22, 34	frec2uz0d 10470	. . . 4 ⊢ (𝜑 → (𝐺‘∅) = 𝐶)
36	25, 33, 35	3eqtr4d 2236	. . 3 ⊢ (𝜑 → (1st ‘(𝑅‘∅)) = (𝐺‘∅))
37	22, 34	frec2uzf1od 10477	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ω–1-1-onto→(ℤ≥‘𝐶))
38		f1of 5500	. . . . . . . . . . 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 5694	. . . . . . . 8 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝐺‘𝑘) ∈ (ℤ≥‘𝐶))
43		peano2uz 9648	. . . . . . . 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 5926	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → ((𝐺‘𝑘)𝐹𝑦) = ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))))
46	45	eleq1d 2262	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(𝑅‘𝑘)) → (((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆 ↔ ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆))
47		oveq1 5925	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐹𝑦) = ((𝐺‘𝑘)𝐹𝑦))
48	47	eleq1d 2262	. . . . . . . . . 10 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((𝐺‘𝑘)𝐹𝑦) ∈ 𝑆))
49	48	ralbidv 2494	. . . . . . . . 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 2576	. . . . . . . . . 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 2869	. . . . . . . 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 10486	. . . . . . . . . . 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 5694	. . . . . . . . 9 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘𝑘) ∈ ((ℤ≥‘𝐶) × 𝑆))
58		xp2nd 6219	. . . . . . . . 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 2869	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘))) ∈ 𝑆)
61		op1stg 6203	. . . . . . 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 6228	. . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . 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 5921	. . . . . . . . . . . . . . . 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 6218	. . . . . . . . . . . . . . . . . 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 6219	. . . . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → (2nd ‘𝑧) ∈ 𝑇)
73		peano2uz 9648	. . . . . . . . . . . . . . . . . . 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 5926	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧)𝐹𝑦) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
76	75	eleq1d 2262	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (2nd ‘𝑧) → (((1st ‘𝑧)𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆))
77		oveq1 5925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥𝐹𝑦) = ((1st ‘𝑧)𝐹𝑦))
78	77	eleq1d 2262	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (1st ‘𝑧) → ((𝑥𝐹𝑦) ∈ 𝑆 ↔ ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆))
79	78	ralbidv 2494	. . . . . . . . . . . . . . . . . . . 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 2869	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ∀𝑦 ∈ 𝑆 ((1st ‘𝑧)𝐹𝑦) ∈ 𝑆)
82	76, 81, 71	rspcdva 2869	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) ∧ 𝑧 ∈ ((ℤ≥‘𝐶) × 𝑆)) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) ∈ 𝑆)
83		opelxpi 4691	. . . . . . . . . . . . . . . . . 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 5925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + 1) = ((1st ‘𝑧) + 1))
86	85, 77	opeq12d 3812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (1st ‘𝑧) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩)
87	75	opeq2d 3811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹𝑦)⟩ = ⟨((1st ‘𝑧) + 1), ((1st ‘𝑧)𝐹(2nd ‘𝑧))⟩)
88		eqid 2193	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)
89	86, 87, 88	ovmpog 6053	. . . . . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . . . 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 2270	. . . . . . . . . . . . . 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 2270	. . . . . . . . . . . . 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 2567	. . . . . . . . . . . 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 9606	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶))
96	22, 95	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶))
97		opelxpi 4691	. . . . . . . . . . . . . 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 6460	. . . . . . . . . . . 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 5555	. . . . . . . . . . 11 ⊢ (𝑅‘suc 𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘suc 𝑘)
103	26	fveq1i 5555	. . . . . . . . . . . 12 ⊢ (𝑅‘𝑘) = (frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)‘𝑘)
104	103	fveq2i 5557	. . . . . . . . . . 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 2251	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩)‘(𝑅‘𝑘)))
106		1st2nd2 6228	. . . . . . . . . . . 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 5558	. . . . . . . . . 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 2226	. . . . . . . . 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 3810	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → ⟨(1st ‘(𝑅‘𝑘)), (2nd ‘(𝑅‘𝑘))⟩ = ⟨(𝐺‘𝑘), (2nd ‘(𝑅‘𝑘))⟩)
112	111	fveq2d 5558	. . . . . . . . 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 2226	. . . . . . . 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 5921	. . . . . . . . 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 3180	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (2nd ‘(𝑅‘𝑘)) ∈ 𝑇)
117		opelxpi 4691	. . . . . . . . . . 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 5925	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥 + 1) = ((𝐺‘𝑘) + 1))
120	119, 47	opeq12d 3812	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐺‘𝑘) → ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩ = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹𝑦)⟩)
121	45	opeq2d 3811	. . . . . . . . . . 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 6053	. . . . . . . . . 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 2240	. . . . . . . 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 2226	. . . . . . 7 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝑅‘suc 𝑘) = ⟨((𝐺‘𝑘) + 1), ((𝐺‘𝑘)𝐹(2nd ‘(𝑅‘𝑘)))⟩)
126	125	fveq2d 5558	. . . . . 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 10472	. . . . . 6 ⊢ (((𝑘 ∈ ω ∧ 𝜑) ∧ (1st ‘(𝑅‘𝑘)) = (𝐺‘𝑘)) → (𝐺‘suc 𝑘) = ((𝐺‘𝑘) + 1))
129	62, 126, 128	3eqtr4d 2236	. . . . 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 4632	. 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 2164  ∀wral 2472  Vcvv 2760   ⊆ wss 3153  ∅c0 3446  ⟨cop 3621   ↦ cmpt 4090  suc csuc 4396  ωcom 4622   × cxp 4657  ⟶wf 5250  –1-1-onto→wf1o 5253  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  1^st c1st 6191  2^nd c2nd 6192  freccfrec 6443  1c1 7873   + caddc 7875  ℤcz 9317  ℤ_≥cuz 9592

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593

This theorem is referenced by:  frecuzrdgdomlem  10488  frecuzrdgfunlem  10490  frecuzrdgsuctlem  10494