Theorem fmucnd 24448

Description: The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)

Hypotheses

Ref	Expression
fmucnd.1	⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
fmucnd.2	⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
fmucnd.3	⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
fmucnd.4	⊢ (𝜑 → 𝐶 ∈ (CauFilu‘𝑈))
fmucnd.5	⊢ 𝐷 = ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎))

Assertion

Ref	Expression
fmucnd	⊢ (𝜑 → 𝐷 ∈ (CauFilu‘𝑉))

Distinct variable groups:   𝐶,𝑎   𝐷,𝑎   𝐹,𝑎   𝑉,𝑎   𝑋,𝑎   𝑌,𝑎   𝜑,𝑎

Allowed substitution hint:   𝑈(𝑎)

Proof of Theorem fmucnd

Dummy variables 𝑐 𝑏 𝑣 𝑟 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmucnd.1	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))
2		fmucnd.4	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (CauFilu‘𝑈))
3		cfilufbas 24445	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu‘𝑈)) → 𝐶 ∈ (fBas‘𝑋))
4	1, 2, 3	syl2anc 595	. . 3 ⊢ (𝜑 → 𝐶 ∈ (fBas‘𝑋))
5		fmucnd.2	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
6		fmucnd.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
7		isucn 24434	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑣 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑣(𝐹‘𝑦)))))
8	7	simprbda 503	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) ∧ 𝐹 ∈ (𝑈 Cnu𝑉)) → 𝐹:𝑋⟶𝑌)
9	1, 5, 6, 8	syl21anc 850	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
10	5	elfvexd 6917	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
11		fmucnd.5	. . . 4 ⊢ 𝐷 = ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎))
12	11	fbasrn 24041	. . 3 ⊢ ((𝐶 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ V) → 𝐷 ∈ (fBas‘𝑌))
13	4, 9, 10, 12	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐷 ∈ (fBas‘𝑌))
14		simplr 780	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝑎 ∈ 𝐶)
15		eqid 2763	. . . . . . . 8 ⊢ (𝐹 “ 𝑎) = (𝐹 “ 𝑎)
16		imaeq2 6058	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → (𝐹 “ 𝑐) = (𝐹 “ 𝑎))
17	16	rspceeqv 3604	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐶 ∧ (𝐹 “ 𝑎) = (𝐹 “ 𝑎)) → ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐))
18	14, 15, 17	sylancl 597	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐))
19		imaexg 7906	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑈 Cnu𝑉) → (𝐹 “ 𝑎) ∈ V)
20		eqid 2763	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
21	20	elrnmpt 5948	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑎) ∈ V → ((𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) ↔ ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐)))
22	6, 19, 21	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) ↔ ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐)))
23	22	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ((𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) ↔ ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐)))
24	18, 23	mpbird 260	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → (𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)))
25		imaeq2 6058	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (𝐹 “ 𝑎) = (𝐹 “ 𝑐))
26	25	cbvmptv 5215	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎)) = (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
27	26	rneqi 5927	. . . . . . 7 ⊢ ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎)) = ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
28	11, 27	eqtri 2786	. . . . . 6 ⊢ 𝐷 = ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
29	24, 28	eleqtrrdi 2874	. . . . 5 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → (𝐹 “ 𝑎) ∈ 𝐷)
30	9	ffnd 6706	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
31	30	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝐹 Fn 𝑋)
32		fbelss 23990	. . . . . . . . 9 ⊢ ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑎 ∈ 𝐶) → 𝑎 ⊆ 𝑋)
33	4, 32	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → 𝑎 ⊆ 𝑋)
34	33	ad4ant13 763	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝑎 ⊆ 𝑋)
35		fmucndlem 24447	. . . . . . 7 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑎 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)))
36	31, 34, 35	syl2anc 595	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)))
37		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
38	37	mpofun 7534	. . . . . . . 8 ⊢ Fun (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
39		funimass2 6619	. . . . . . . 8 ⊢ ((Fun (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
40	38, 39	mpan 702	. . . . . . 7 ⊢ ((𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
41	40	adantl 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
42	36, 41	eqsstrrd 3972	. . . . 5 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)) ⊆ 𝑣)
43		id 23	. . . . . . . 8 ⊢ (𝑏 = (𝐹 “ 𝑎) → 𝑏 = (𝐹 “ 𝑎))
44	43	sqxpeqd 5693	. . . . . . 7 ⊢ (𝑏 = (𝐹 “ 𝑎) → (𝑏 × 𝑏) = ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)))
45	44	sseq1d 3968	. . . . . 6 ⊢ (𝑏 = (𝐹 “ 𝑎) → ((𝑏 × 𝑏) ⊆ 𝑣 ↔ ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)) ⊆ 𝑣))
46	45	rspcev 3581	. . . . 5 ⊢ (((𝐹 “ 𝑎) ∈ 𝐷 ∧ ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)) ⊆ 𝑣) → ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
47	29, 42, 46	syl2anc 595	. . . 4 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
48	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
49	2	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ (CauFilu‘𝑈))
50	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑉 ∈ (UnifOn‘𝑌))
51	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐹 ∈ (𝑈 Cnu𝑉))
52		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
53		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑠⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩
54		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑡⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩
55		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩
56		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑦⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩
57		simpl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → 𝑥 = 𝑠)
58	57	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝐹‘𝑥) = (𝐹‘𝑠))
59		simpr 489	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → 𝑦 = 𝑡)
60	59	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝐹‘𝑦) = (𝐹‘𝑡))
61	58, 60	opeq12d 4846	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩)
62	53, 54, 55, 56, 61	cbvmpo 7504	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑠 ∈ 𝑋, 𝑡 ∈ 𝑋 ↦ ⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩)
63	48, 50, 51, 52, 62	ucnprima 24438	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣) ∈ 𝑈)
64		cfiluexsm 24446	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu‘𝑈) ∧ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣) ∈ 𝑈) → ∃𝑎 ∈ 𝐶 (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣))
65	48, 49, 63, 64	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ∃𝑎 ∈ 𝐶 (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣))
66	47, 65	r19.29a 3173	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
67	66	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
68		iscfilu 24444	. . 3 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → (𝐷 ∈ (CauFilu‘𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
69	5, 68	syl 18	. 2 ⊢ (𝜑 → (𝐷 ∈ (CauFilu‘𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
70	13, 67, 69	mpbir2and 725	1 ⊢ (𝜑 → 𝐷 ∈ (CauFilu‘𝑉))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  ^◡ccnv 5660  ran crn 5662   “ cima 5664  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  fBascfbas 21510  UnifOncust 24357   Cn_ucucn 24431  CauFil_uccfilu 24442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-fbas 21519  df-ust 24358  df-ucn 24432  df-cfilu 24443

This theorem is referenced by:  ucnextcn  24460