Theorem cnpflf2 24024

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 23274	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1117	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1167	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		simpl1 1190	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5		simpl3 1192	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
6		neiflim 23998	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7		cnpflf2.3	. . . . . . 7 ⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})
8	7	oveq2i 7442	. . . . . 6 ⊢ (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
9	6, 8	eleqtrrdi 2850	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
10	4, 5, 9	syl2anc 584	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11		simpr 484	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12		cnpflfi 24023	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
13	10, 11, 12	syl2anc 584	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
14	3, 13	jca 511	. 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 1190	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16		topontop 22935	. . . . . . . . . . . 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 1192	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
19		toponuni 22936	. . . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ ∪ 𝐽)
22	7	eleq2i 2831	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐿 ↔ 𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23		eqid 2735	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	isneip 23129	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
25	22, 24	bitrid 283	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ∪ 𝐽) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
26	17, 21, 25	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (𝑧 ∈ 𝐿 ↔ (𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))))
27		sstr2 4002	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧) → ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝐹 “ 𝑣) ⊆ 𝑢))
28		imass2 6123	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧))
29	27, 28	syl11 33	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝑣 ⊆ 𝑧 → (𝐹 “ 𝑣) ⊆ 𝑢))
30	29	anim2d 612	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ 𝑧) ⊆ 𝑢 → ((𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) → (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
31	30	reximdv 3168	. . . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ ∪ 𝐽 ∧ ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧)) → ((𝐹 “ 𝑧) ⊆ 𝑢 → ∃𝑣 ∈ 𝐽 (𝐴 ∈ 𝑣 ∧ (𝐹 “ 𝑣) ⊆ 𝑢)))
34	26, 33	biimtrdi 253	. . . . . . . . 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 3151	. . . . . . . 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 3167	. . . . . 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 484	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌)
39	37, 38	jctild 525	. . . . 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 490	. . . 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 1191	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌))
42	18	snssd 4814	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
43	18	snn0d 4780	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
44		neifil 23904	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
45	15, 42, 43, 44	syl3anc 1370	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	7, 45	eqeltrid 2843	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐿 ∈ (Fil‘𝑋))
47		isflf 24017	. . . . 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 1370	. . . 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 23261	. . . . 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 480	. . . 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 294	. . 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 454	. 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 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ cuni 4912   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932  neicnei 23121   CnP ccnp 23249  Filcfil 23869   fLim cflim 23958   fLimf cflf 23959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-cnp 23252  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964

This theorem is referenced by:  cnpflf  24025