Theorem cnpflf2 24166

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 23416	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1136	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1187	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		simpl1 1210	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5		simpl3 1212	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
6		neiflim 24140	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7		cnpflf2.3	. . . . . . 7 ⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})
8	7	oveq2i 7421	. . . . . 6 ⊢ (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
9	6, 8	eleqtrrdi 2874	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
10	4, 5, 9	syl2anc 595	. . . 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 24165	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
13	10, 11, 12	syl2anc 595	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
14	3, 13	jca 520	. 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 1210	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16		topontop 23079	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
17	15, 16	syl 18	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ Top)
18		simpl3 1212	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
19		toponuni 23080	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
20	15, 19	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 = ∪ 𝐽)
21	18, 20	eleqtrd 2865	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ ∪ 𝐽)
22	7	eleq2i 2855	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐿 ↔ 𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23		eqid 2763	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	isneip 23271	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
25	22, 24	bitrid 286	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
26	17, 21, 25	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
27		sstr2 3944	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧) → ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝐹 “ 𝑣) ⊆ 𝑢))
28		imass2 6104	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧))
29	27, 28	syl11 34	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ 𝑢))
30	29	anim2d 623	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → ((𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
31	30	reximdv 3180	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → (∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
32	31	com12 33	. . . . . . . . . . 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	biimtrdi 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 3164	. . . . . . . 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 58	. . . . . . 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 3179	. . . . . 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 534	. . . . 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 1211	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌))
42	18	snssd 4752	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
43	18	snn0d 4741	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
44		neifil 24046	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
45	15, 42, 43, 44	syl3anc 1398	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	7, 45	eqeltrid 2867	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐿 ∈ (Fil‘𝑋))
47		isflf 24159	. . . . 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))))
48	41, 46, 38, 47	syl3anc 1398	. . . 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))))
49		iscnp 23403	. . . . 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)))))
50	49	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)))))
51	40, 48, 50	3imtr4d 297	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
52	51	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))
53	14, 52	impbida 812	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  ∅c0 4286  {csn 4589  ∪ cuni 4872   “ cima 5664  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Topctop 23059  TopOnctopon 23076  neicnei 23263   CnP ccnp 23391  Filcfil 24011   fLim cflim 24100   fLimf cflf 24101

This proof depends on 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 proof 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-reu 3370  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-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-fbas 21528  df-fg 21529  df-top 23060  df-topon 23077  df-ntr 23186  df-nei 23264  df-cnp 23394  df-fil 24012  df-fm 24104  df-flim 24105  df-flf 24106

This theorem is used by:  cnpflf  24167