Theorem cnpflf2 21804

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 21054	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1265	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1219	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		simpl1 1064	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5		simpl3 1066	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
6		neiflim 21778	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7		cnpflf2.3	. . . . . . 7 ⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})
8	7	oveq2i 6661	. . . . . 6 ⊢ (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
9	6, 8	syl6eleqr 2712	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
10	4, 5, 9	syl2anc 693	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11		simpr 477	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12		cnpflfi 21803	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
13	10, 11, 12	syl2anc 693	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
14	3, 13	jca 554	. 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 1064	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16		topontop 20718	. . . . . . . . . . . 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 1066	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
19		toponuni 20719	. . . . . . . . . . . . 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 2703	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ ∪ 𝐽)
22	7	eleq2i 2693	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐿 ↔ 𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	isneip 20909	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
25	22, 24	syl5bb 272	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
26	17, 21, 25	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
27		sstr2 3610	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧) → ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝐹 “ 𝑣) ⊆ 𝑢))
28		imass2 5501	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧))
29	27, 28	syl11 33	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ 𝑢))
30	29	anim2d 589	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → ((𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
31	30	reximdv 3016	. . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧)) → ((𝐹 “ 𝑧) ⊆ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
34	26, 33	syl6bi 243	. . . . . . . . 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 3030	. . . . . . . 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 2963	. . . . . 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 477	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌)
39	37, 38	jctild 566	. . . . 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 483	. . . 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 1065	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌))
42	18	snssd 4340	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
43		snnzg 4308	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → {𝐴} ≠ ∅)
44	18, 43	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
45		neifil 21684	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	15, 42, 44, 45	syl3anc 1326	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
47	7, 46	syl5eqel 2705	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐿 ∈ (Fil‘𝑋))
48		isflf 21797	. . . . 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 1326	. . . 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 21041	. . . . 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 481	. . . 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 283	. . 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 649	. 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 877	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Topctop 20698  TopOnctopon 20715  neicnei 20901   CnP ccnp 21029  Filcfil 21649   fLim cflim 21738   fLimf cflf 21739

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-reu 2919  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-f1 5893  df-fo 5894  df-f1o 5895  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-fg 19744  df-top 20699  df-topon 20716  df-ntr 20824  df-nei 20902  df-cnp 21032  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744

This theorem is referenced by:  cnpflf  21805