Theorem cnpflf2 22701

Description: 𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹‘𝐴). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
cnpflf2.3	⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})

Assertion

Ref	Expression
cnpflf2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))

Proof of Theorem cnpflf2

Dummy variables 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnpf2 21951	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1116	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1166	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		simpl1 1189	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5		simpl3 1191	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
6		neiflim 22675	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7		cnpflf2.3	. . . . . . 7 ⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})
8	7	oveq2i 7162	. . . . . 6 ⊢ (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
9	6, 8	eleqtrrdi 2864	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
10	4, 5, 9	syl2anc 588	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11		simpr 489	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12		cnpflfi 22700	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
13	10, 11, 12	syl2anc 588	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
14	3, 13	jca 516	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)))
15		simpl1 1189	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16		topontop 21614	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ Top)
18		simpl3 1191	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
19		toponuni 21615	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
20	15, 19	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 = ∪ 𝐽)
21	18, 20	eleqtrd 2855	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ ∪ 𝐽)
22	7	eleq2i 2844	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐿 ↔ 𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23		eqid 2759	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	isneip 21806	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
25	22, 24	syl5bb 286	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
26	17, 21, 25	syl2anc 588	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
27		sstr2 3900	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧) → ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝐹 “ 𝑣) ⊆ 𝑢))
28		imass2 5938	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧))
29	27, 28	syl11 33	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ 𝑢))
30	29	anim2d 615	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → ((𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
31	30	reximdv 3198	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → (∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
32	31	com12 32	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → ((𝐹 “ 𝑧) ⊆ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
33	32	adantl 486	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧)) → ((𝐹 “ 𝑧) ⊆ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
34	26, 33	syl6bi 256	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝑧 ∈ 𝐿 → ((𝐹 “ 𝑧) ⊆ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
35	34	rexlimdv 3208	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
36	35	imim2d 57	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (((𝐹‘𝐴) ∈ 𝑢 → ∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢) → ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
37	36	ralimdv 3110	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢) → ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢))))
38		simpr 489	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌)
39	37, 38	jctild 530	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))))
40	39	adantld 495	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (((𝐹‘𝐴) ∈ 𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢)) → (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))))
41		simpl2 1190	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌))
42	18	snssd 4700	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
43		snnzg 4668	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
44	18, 43	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
45		neifil 22581	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	15, 42, 44, 45	syl3anc 1369	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
47	7, 46	eqeltrid 2857	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐿 ∈ (Fil‘𝑋))
48		isflf 22694	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹‘𝐴) ∈ 𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢))))
49	41, 47, 38, 48	syl3anc 1369	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹‘𝐴) ∈ 𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑧 ∈ 𝐿 (𝐹 “ 𝑧) ⊆ 𝑢))))
50		iscnp 21938	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))))
51	50	adantr 485	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))))
52	40, 49, 51	3imtr4d 298	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
53	52	impr 459	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
54	14, 53	impbida 801	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072   ⊆ wss 3859  ∅c0 4226  {csn 4523  ∪ cuni 4799   “ cima 5528  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  Topctop 21594  TopOnctopon 21611  neicnei 21798   CnP ccnp 21926  Filcfil 22546   fLim cflim 22635   fLimf cflf 22636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-map 8419  df-fbas 20164  df-fg 20165  df-top 21595  df-topon 21612  df-ntr 21721  df-nei 21799  df-cnp 21929  df-fil 22547  df-fm 22639  df-flim 22640  df-flf 22641

This theorem is referenced by:  cnpflf  22702