Theorem fmucnd 22096

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 22093	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐶 ∈ (CauFilu‘𝑈)) → 𝐶 ∈ (fBas‘𝑋))
4	1, 2, 3	syl2anc 693	. . 3 ⊢ (𝜑 → 𝐶 ∈ (fBas‘𝑋))
5		fmucnd.2	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))
6		fmucnd.3	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))
7		isucn 22082	. . . . 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 653	. . . 4 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) ∧ 𝐹 ∈ (𝑈 Cnu𝑉)) → 𝐹:𝑋⟶𝑌)
9	1, 5, 6, 8	syl21anc 1325	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
10	5	elfvexd 6222	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
11		fmucnd.5	. . . 4 ⊢ 𝐷 = ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎))
12	11	fbasrn 21688	. . 3 ⊢ ((𝐶 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ V) → 𝐷 ∈ (fBas‘𝑌))
13	4, 9, 10, 12	syl3anc 1326	. 2 ⊢ (𝜑 → 𝐷 ∈ (fBas‘𝑌))
14		simplr 792	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝑎 ∈ 𝐶)
15		eqid 2622	. . . . . . . 8 ⊢ (𝐹 “ 𝑎) = (𝐹 “ 𝑎)
16		imaeq2 5462	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (𝐹 “ 𝑐) = (𝐹 “ 𝑎))
17	16	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → ((𝐹 “ 𝑎) = (𝐹 “ 𝑐) ↔ (𝐹 “ 𝑎) = (𝐹 “ 𝑎)))
18	17	rspcev 3309	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐶 ∧ (𝐹 “ 𝑎) = (𝐹 “ 𝑎)) → ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐))
19	14, 15, 18	sylancl 694	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐))
20		imaexg 7103	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑈 Cnu𝑉) → (𝐹 “ 𝑎) ∈ V)
21		eqid 2622	. . . . . . . . . 10 ⊢ (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) = (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
22	21	elrnmpt 5372	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑎) ∈ V → ((𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) ↔ ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐)))
23	6, 20, 22	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)) ↔ ∃𝑐 ∈ 𝐶 (𝐹 “ 𝑎) = (𝐹 “ 𝑐)))
24	23	ad3antrrr 766	. . . . . . 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)))
25	19, 24	mpbird 247	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → (𝐹 “ 𝑎) ∈ ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐)))
26		imaeq2 5462	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → (𝐹 “ 𝑎) = (𝐹 “ 𝑐))
27	26	cbvmptv 4750	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎)) = (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
28	27	rneqi 5352	. . . . . . 7 ⊢ ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎)) = ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
29	11, 28	eqtri 2644	. . . . . 6 ⊢ 𝐷 = ran (𝑐 ∈ 𝐶 ↦ (𝐹 “ 𝑐))
30	25, 29	syl6eleqr 2712	. . . . 5 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → (𝐹 “ 𝑎) ∈ 𝐷)
31		ffn 6045	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋)
32	9, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
33	32	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝐹 Fn 𝑋)
34		simplll 798	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝜑)
35		fbelss 21637	. . . . . . . . 9 ⊢ ((𝐶 ∈ (fBas‘𝑋) ∧ 𝑎 ∈ 𝐶) → 𝑎 ⊆ 𝑋)
36	4, 35	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶) → 𝑎 ⊆ 𝑋)
37	34, 14, 36	syl2anc 693	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → 𝑎 ⊆ 𝑋)
38		fmucndlem 22095	. . . . . . 7 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑎 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) = ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)))
39	33, 37, 38	syl2anc 693	. . . . . 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)))
40		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
41	40	mpt2fun 6762	. . . . . . . 8 ⊢ Fun (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)
42		funimass2 5972	. . . . . . . 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)
43	41, 42	mpan 706	. . . . . . 7 ⊢ ((𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝑎 × 𝑎)) ⊆ 𝑣)
44	43	adantl 482	. . . . . 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)
45	39, 44	eqsstr3d 3640	. . . . 5 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)) ⊆ 𝑣)
46		id 22	. . . . . . . 8 ⊢ (𝑏 = (𝐹 “ 𝑎) → 𝑏 = (𝐹 “ 𝑎))
47	46	sqxpeqd 5141	. . . . . . 7 ⊢ (𝑏 = (𝐹 “ 𝑎) → (𝑏 × 𝑏) = ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)))
48	47	sseq1d 3632	. . . . . 6 ⊢ (𝑏 = (𝐹 “ 𝑎) → ((𝑏 × 𝑏) ⊆ 𝑣 ↔ ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)) ⊆ 𝑣))
49	48	rspcev 3309	. . . . 5 ⊢ (((𝐹 “ 𝑎) ∈ 𝐷 ∧ ((𝐹 “ 𝑎) × (𝐹 “ 𝑎)) ⊆ 𝑣) → ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
50	30, 45, 49	syl2anc 693	. . . 4 ⊢ ((((𝜑 ∧ 𝑣 ∈ 𝑉) ∧ 𝑎 ∈ 𝐶) ∧ (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣)) → ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
51	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑈 ∈ (UnifOn‘𝑋))
52	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ (CauFilu‘𝑈))
53	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑉 ∈ (UnifOn‘𝑌))
54	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐹 ∈ (𝑈 Cnu𝑉))
55		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
56		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑠⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩
57		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑡⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩
58		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩
59		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑦⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩
60		simpl 473	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → 𝑥 = 𝑠)
61	60	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝐹‘𝑥) = (𝐹‘𝑠))
62		simpr 477	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → 𝑦 = 𝑡)
63	62	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → (𝐹‘𝑦) = (𝐹‘𝑡))
64	61, 63	opeq12d 4410	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 𝑡) → ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩ = ⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩)
65	56, 57, 58, 59, 64	cbvmpt2 6734	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) = (𝑠 ∈ 𝑋, 𝑡 ∈ 𝑋 ↦ ⟨(𝐹‘𝑠), (𝐹‘𝑡)⟩)
66	51, 53, 54, 55, 65	ucnprima 22086	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣) ∈ 𝑈)
67		cfiluexsm 22094	. . . . 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))
68	51, 52, 66, 67	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ∃𝑎 ∈ 𝐶 (𝑎 × 𝑎) ⊆ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ 𝑣))
69	50, 68	r19.29a 3078	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
70	69	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)
71		iscfilu 22092	. . 3 ⊢ (𝑉 ∈ (UnifOn‘𝑌) → (𝐷 ∈ (CauFilu‘𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
72	5, 71	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (CauFilu‘𝑉) ↔ (𝐷 ∈ (fBas‘𝑌) ∧ ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝐷 (𝑏 × 𝑏) ⊆ 𝑣)))
73	13, 70, 72	mpbir2and 957	1 ⊢ (𝜑 → 𝐷 ∈ (CauFilu‘𝑉))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  fBascfbas 19734  UnifOncust 22003   Cn_ucucn 22079  CauFil_uccfilu 22090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-fbas 19743  df-ust 22004  df-ucn 22080  df-cfilu 22091

This theorem is referenced by:  ucnextcn  22108