Theorem ennnfonelemp1 12977

Description: Lemma for ennnfone 12996. Value of 𝐻 at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)

Hypotheses

Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)
ennnfonelemp1.p	⊢ (𝜑 → 𝑃 ∈ ℕ0)

Assertion

Ref	Expression
ennnfonelemp1	⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))

Distinct variable groups:   𝐴,𝑗,𝑥,𝑦   𝑥,𝐹,𝑦   𝑗,𝐺   𝑥,𝐻,𝑦   𝑗,𝐽   𝑥,𝑁,𝑦   𝑃,𝑗,𝑥,𝑦   𝜑,𝑗,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑘,𝑛)   𝐴(𝑘,𝑛)   𝑃(𝑘,𝑛)   𝐹(𝑗,𝑘,𝑛)   𝐺(𝑥,𝑦,𝑘,𝑛)   𝐻(𝑗,𝑘,𝑛)   𝐽(𝑥,𝑦,𝑘,𝑛)   𝑁(𝑗,𝑘,𝑛)

Proof of Theorem ennnfonelemp1

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemp1.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ0)
2		nn0uz 9757	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2322	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘0))
4		ennnfonelemh.dceq	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
5		ennnfonelemh.f	. . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
6		ennnfonelemh.ne	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
7		ennnfonelemh.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
8		ennnfonelemh.n	. . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
9		ennnfonelemh.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
10		ennnfonelemh.h	. . . . 5 ⊢ 𝐻 = seq0(𝐺, 𝐽)
11	4, 5, 6, 7, 8, 9, 10	ennnfonelemj0 12972	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
12	4, 5, 6, 7, 8, 9, 10	ennnfonelemg 12974	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
13	4, 5, 6, 7, 8, 9, 10	ennnfonelemjn 12973	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
14	3, 11, 12, 13	seqp1cd 10692	. . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘(𝑃 + 1)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
15	10	fveq1i 5628	. . . 4 ⊢ (𝐻‘(𝑃 + 1)) = (seq0(𝐺, 𝐽)‘(𝑃 + 1))
16	15	a1i 9	. . 3 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = (seq0(𝐺, 𝐽)‘(𝑃 + 1)))
17	10	fveq1i 5628	. . . . 5 ⊢ (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃)
18	17	a1i 9	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃))
19		eqeq1 2236	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (𝑥 = 0 ↔ (𝑃 + 1) = 0))
20		fvoveq1 6024	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘((𝑃 + 1) − 1)))
21	19, 20	ifbieq2d 3627	. . . . . 6 ⊢ (𝑥 = (𝑃 + 1) → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))))
22		peano2nn0 9409	. . . . . . 7 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ∈ ℕ0)
23	1, 22	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑃 + 1) ∈ ℕ0)
24		nn0p1gt0 9398	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0 → 0 < (𝑃 + 1))
25	24	gt0ne0d 8659	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ≠ 0)
26	25	neneqd 2421	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ¬ (𝑃 + 1) = 0)
27	26	iffalsed 3612	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘((𝑃 + 1) − 1)))
28		nn0cn 9379	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 𝑃 ∈ ℂ)
29		1cnd 8162	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 1 ∈ ℂ)
30	28, 29	pncand 8458	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ((𝑃 + 1) − 1) = 𝑃)
31	30	fveq2d 5631	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘((𝑃 + 1) − 1)) = (◡𝑁‘𝑃))
32	27, 31	eqtrd 2262	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘𝑃))
33	8	frechashgf1o 10650	. . . . . . . . . . 11 ⊢ 𝑁:ω–1-1-onto→ℕ0
34		f1ocnv 5585	. . . . . . . . . . 11 ⊢ (𝑁:ω–1-1-onto→ℕ0 → ◡𝑁:ℕ0–1-1-onto→ω)
35	33, 34	ax-mp 5	. . . . . . . . . 10 ⊢ ◡𝑁:ℕ0–1-1-onto→ω
36		f1of 5572	. . . . . . . . . 10 ⊢ (◡𝑁:ℕ0–1-1-onto→ω → ◡𝑁:ℕ0⟶ω)
37	35, 36	mp1i 10	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → ◡𝑁:ℕ0⟶ω)
38		id 19	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → 𝑃 ∈ ℕ0)
39	37, 38	ffvelcdmd 5771	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘𝑃) ∈ ω)
40	32, 39	eqeltrd 2306	. . . . . . 7 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) ∈ ω)
41	1, 40	syl 14	. . . . . 6 ⊢ (𝜑 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) ∈ ω)
42	9, 21, 23, 41	fvmptd3 5728	. . . . 5 ⊢ (𝜑 → (𝐽‘(𝑃 + 1)) = if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))))
43	1, 32	syl 14	. . . . 5 ⊢ (𝜑 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘𝑃))
44	42, 43	eqtr2d 2263	. . . 4 ⊢ (𝜑 → (◡𝑁‘𝑃) = (𝐽‘(𝑃 + 1)))
45	18, 44	oveq12d 6019	. . 3 ⊢ (𝜑 → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
46	14, 16, 45	3eqtr4d 2272	. 2 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)))
47	4, 5, 6, 7, 8, 9, 10	ennnfonelemh 12975	. . . 4 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
48	47, 1	ffvelcdmd 5771	. . 3 ⊢ (𝜑 → (𝐻‘𝑃) ∈ (𝐴 ↑pm ω))
49	1, 39	syl 14	. . 3 ⊢ (𝜑 → (◡𝑁‘𝑃) ∈ ω)
50	48	elexd 2813	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) ∈ V)
51		dmexg 4988	. . . . . . . 8 ⊢ ((𝐻‘𝑃) ∈ V → dom (𝐻‘𝑃) ∈ V)
52	50, 51	syl 14	. . . . . . 7 ⊢ (𝜑 → dom (𝐻‘𝑃) ∈ V)
53		fof 5548	. . . . . . . . 9 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
54	5, 53	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω⟶𝐴)
55	54, 49	ffvelcdmd 5771	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴)
56		opexg 4314	. . . . . . 7 ⊢ ((dom (𝐻‘𝑃) ∈ V ∧ (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴) → ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V)
57	52, 55, 56	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V)
58		snexg 4268	. . . . . 6 ⊢ (⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩ ∈ V → {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩} ∈ V)
59	57, 58	syl 14	. . . . 5 ⊢ (𝜑 → {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩} ∈ V)
60		unexg 4534	. . . . 5 ⊢ (((𝐻‘𝑃) ∈ V ∧ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩} ∈ V) → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}) ∈ V)
61	50, 59, 60	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}) ∈ V)
62	4, 5, 49	ennnfonelemdc 12970	. . . 4 ⊢ (𝜑 → DECID (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)))
63	50, 61, 62	ifcldcd 3640	. . 3 ⊢ (𝜑 → if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})) ∈ V)
64		id 19	. . . . 5 ⊢ (𝑥 = (𝐻‘𝑃) → 𝑥 = (𝐻‘𝑃))
65		dmeq 4923	. . . . . . . 8 ⊢ (𝑥 = (𝐻‘𝑃) → dom 𝑥 = dom (𝐻‘𝑃))
66	65	opeq1d 3863	. . . . . . 7 ⊢ (𝑥 = (𝐻‘𝑃) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩)
67	66	sneqd 3679	. . . . . 6 ⊢ (𝑥 = (𝐻‘𝑃) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})
68	64, 67	uneq12d 3359	. . . . 5 ⊢ (𝑥 = (𝐻‘𝑃) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}))
69	64, 68	ifeq12d 3622	. . . 4 ⊢ (𝑥 = (𝐻‘𝑃) → if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})) = if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})))
70		fveq2 5627	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹‘𝑦) = (𝐹‘(◡𝑁‘𝑃)))
71		imaeq2 5064	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝑁‘𝑃)))
72	70, 71	eleq12d 2300	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐹‘𝑦) ∈ (𝐹 “ 𝑦) ↔ (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃))))
73	70	opeq2d 3864	. . . . . . 7 ⊢ (𝑦 = (◡𝑁‘𝑃) → ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩)
74	73	sneqd 3679	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})
75	74	uneq2d 3358	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}))
76	72, 75	ifbieq2d 3627	. . . 4 ⊢ (𝑦 = (◡𝑁‘𝑃) → if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
77	69, 76, 7	ovmpog 6139	. . 3 ⊢ (((𝐻‘𝑃) ∈ (𝐴 ↑pm ω) ∧ (◡𝑁‘𝑃) ∈ ω ∧ if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})) ∈ V) → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
78	48, 49, 63, 77	syl3anc 1271	. 2 ⊢ (𝜑 → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))
79	46, 78	eqtrd 2262	1 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = if((𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)), (𝐻‘𝑃), ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})))

Syntax hints:   → wi 4  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  ∃wrex 2509  {crab 2512  Vcvv 2799   ∪ cun 3195  ∅c0 3491  ifcif 3602  {csn 3666  ⟨cop 3669   ↦ cmpt 4145  suc csuc 4456  ωcom 4682  ^◡ccnv 4718  dom cdm 4719   “ cima 4722  ⟶wf 5314  –onto→wfo 5316  –1-1-onto→wf1o 5317  ‘cfv 5318  (class class class)co 6001   ∈ cmpo 6003  freccfrec 6536   ↑_pm cpm 6796  0cc0 7999  1c1 8000   + caddc 8002   − cmin 8317  ℕ₀cn0 9369  ℤcz 9446  ℤ_≥cuz 9722  seqcseq 10669

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pm 6798  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-seqfrec 10670

This theorem is referenced by:  ennnfonelem1  12978  ennnfonelemhdmp1  12980  ennnfonelemss  12981  ennnfonelemkh  12983  ennnfonelemhf1o  12984