Theorem ennnfonelemp1 13157

Description: Lemma for ennnfone 13176. 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 9889	. . . . 5 ⊢ ℕ0 = (ℤ≥‘0)
3	1, 2	eleqtrdi 2325	. . . 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 13152	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω})
12	4, 5, 6, 7, 8, 9, 10	ennnfonelemg 13154	. . . 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 13153	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
14	3, 11, 12, 13	seqp1cd 10832	. . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘(𝑃 + 1)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
15	10	fveq1i 5671	. . . 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 5671	. . . . 5 ⊢ (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃)
18	17	a1i 9	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) = (seq0(𝐺, 𝐽)‘𝑃))
19		eqeq1 2239	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (𝑥 = 0 ↔ (𝑃 + 1) = 0))
20		fvoveq1 6073	. . . . . . 7 ⊢ (𝑥 = (𝑃 + 1) → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘((𝑃 + 1) − 1)))
21	19, 20	ifbieq2d 3647	. . . . . 6 ⊢ (𝑥 = (𝑃 + 1) → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))))
22		peano2nn0 9536	. . . . . . 7 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ∈ ℕ0)
23	1, 22	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑃 + 1) ∈ ℕ0)
24		nn0p1gt0 9525	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ0 → 0 < (𝑃 + 1))
25	24	gt0ne0d 8786	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → (𝑃 + 1) ≠ 0)
26	25	neneqd 2433	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ¬ (𝑃 + 1) = 0)
27	26	iffalsed 3632	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘((𝑃 + 1) − 1)))
28		nn0cn 9506	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 𝑃 ∈ ℂ)
29		1cnd 8290	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ0 → 1 ∈ ℂ)
30	28, 29	pncand 8585	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ0 → ((𝑃 + 1) − 1) = 𝑃)
31	30	fveq2d 5674	. . . . . . . . 9 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘((𝑃 + 1) − 1)) = (◡𝑁‘𝑃))
32	27, 31	eqtrd 2265	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → if((𝑃 + 1) = 0, ∅, (◡𝑁‘((𝑃 + 1) − 1))) = (◡𝑁‘𝑃))
33	8	frechashgf1o 10790	. . . . . . . . . . 11 ⊢ 𝑁:ω–1-1-onto→ℕ0
34		f1ocnv 5627	. . . . . . . . . . 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 5614	. . . . . . . . . 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 5813	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ0 → (◡𝑁‘𝑃) ∈ ω)
40	32, 39	eqeltrd 2309	. . . . . . 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 5771	. . . . 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 2266	. . . 4 ⊢ (𝜑 → (◡𝑁‘𝑃) = (𝐽‘(𝑃 + 1)))
45	18, 44	oveq12d 6068	. . 3 ⊢ (𝜑 → ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)) = ((seq0(𝐺, 𝐽)‘𝑃)𝐺(𝐽‘(𝑃 + 1))))
46	14, 16, 45	3eqtr4d 2275	. 2 ⊢ (𝜑 → (𝐻‘(𝑃 + 1)) = ((𝐻‘𝑃)𝐺(◡𝑁‘𝑃)))
47	4, 5, 6, 7, 8, 9, 10	ennnfonelemh 13155	. . . 4 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
48	47, 1	ffvelcdmd 5813	. . 3 ⊢ (𝜑 → (𝐻‘𝑃) ∈ (𝐴 ↑pm ω))
49	1, 39	syl 14	. . 3 ⊢ (𝜑 → (◡𝑁‘𝑃) ∈ ω)
50	48	elexd 2827	. . . 4 ⊢ (𝜑 → (𝐻‘𝑃) ∈ V)
51		dmexg 5021	. . . . . . . 8 ⊢ ((𝐻‘𝑃) ∈ V → dom (𝐻‘𝑃) ∈ V)
52	50, 51	syl 14	. . . . . . 7 ⊢ (𝜑 → dom (𝐻‘𝑃) ∈ V)
53		fof 5590	. . . . . . . . 9 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
54	5, 53	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω⟶𝐴)
55	54, 49	ffvelcdmd 5813	. . . . . . 7 ⊢ (𝜑 → (𝐹‘(◡𝑁‘𝑃)) ∈ 𝐴)
56		opexg 4344	. . . . . . 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 4297	. . . . . 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 4564	. . . . 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 13150	. . . 4 ⊢ (𝜑 → DECID (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃)))
63	50, 61, 62	ifcldcd 3660	. . 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 4956	. . . . . . . 8 ⊢ (𝑥 = (𝐻‘𝑃) → dom 𝑥 = dom (𝐻‘𝑃))
66	65	opeq1d 3889	. . . . . . 7 ⊢ (𝑥 = (𝐻‘𝑃) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩)
67	66	sneqd 3702	. . . . . 6 ⊢ (𝑥 = (𝐻‘𝑃) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩})
68	64, 67	uneq12d 3374	. . . . 5 ⊢ (𝑥 = (𝐻‘𝑃) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}))
69	64, 68	ifeq12d 3642	. . . 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 5670	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹‘𝑦) = (𝐹‘(◡𝑁‘𝑃)))
71		imaeq2 5097	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → (𝐹 “ 𝑦) = (𝐹 “ (◡𝑁‘𝑃)))
72	70, 71	eleq12d 2303	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐹‘𝑦) ∈ (𝐹 “ 𝑦) ↔ (𝐹‘(◡𝑁‘𝑃)) ∈ (𝐹 “ (◡𝑁‘𝑃))))
73	70	opeq2d 3890	. . . . . . 7 ⊢ (𝑦 = (◡𝑁‘𝑃) → ⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩ = ⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩)
74	73	sneqd 3702	. . . . . 6 ⊢ (𝑦 = (◡𝑁‘𝑃) → {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩} = {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩})
75	74	uneq2d 3373	. . . . 5 ⊢ (𝑦 = (◡𝑁‘𝑃) → ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘𝑦)⟩}) = ((𝐻‘𝑃) ∪ {⟨dom (𝐻‘𝑃), (𝐹‘(◡𝑁‘𝑃))⟩}))
76	72, 75	ifbieq2d 3647	. . . 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 6188	. . 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 1274	. 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 2265	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 842   = wceq 1398   ∈ wcel 2203   ≠ wne 2412  ∀wral 2520  ∃wrex 2521  {crab 2524  Vcvv 2813   ∪ cun 3209  ∅c0 3508  ifcif 3620  {csn 3689  ⟨cop 3692   ↦ cmpt 4171  suc csuc 4486  ωcom 4712  ^◡ccnv 4748  dom cdm 4749   “ cima 4752  ⟶wf 5348  –onto→wfo 5350  –1-1-onto→wf1o 5351  ‘cfv 5352  (class class class)co 6050   ∈ cmpo 6052  freccfrec 6621   ↑_pm cpm 6883  0cc0 8127  1c1 8128   + caddc 8130   − cmin 8444  ℕ₀cn0 9496  ℤcz 9577  ℤ_≥cuz 9853  seqcseq 10809

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pm 6885  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810

This theorem is referenced by:  ennnfonelem1  13158  ennnfonelemhdmp1  13160  ennnfonelemss  13161  ennnfonelemkh  13163  ennnfonelemhf1o  13164