Theorem cnpflf2 23915

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 23165	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
2	1	3expa 1118	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
3	2	3adantl3 1169	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶𝑌)
4		simpl1 1192	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5		simpl3 1194	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
6		neiflim 23889	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7		cnpflf2.3	. . . . . . 7 ⊢ 𝐿 = ((nei‘𝐽)‘{𝐴})
8	7	oveq2i 7357	. . . . . 6 ⊢ (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
9	6, 8	eleqtrrdi 2842	. . . . 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 23914	. . . 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 1192	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16		topontop 22828	. . . . . . . . . . . 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 1194	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ 𝑋)
19		toponuni 22829	. . . . . . . . . . . . 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 2833	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐴 ∈ ∪ 𝐽)
22	7	eleq2i 2823	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐿 ↔ 𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23		eqid 2731	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
24	23	isneip 23020	. . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑣) ⊆ (𝐹 “ 𝑧) → ((𝐹 “ 𝑧) ⊆ 𝑢 → (𝐹 “ 𝑣) ⊆ 𝑢))
28		imass2 6050	. . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . 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 3131	. . . . . . . 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 3146	. . . . . 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 1193	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌))
42	18	snssd 4758	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ⊆ 𝑋)
43	18	snn0d 4725	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → {𝐴} ≠ ∅)
44		neifil 23795	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
45	15, 42, 43, 44	syl3anc 1373	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
46	7, 45	eqeltrid 2835	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐿 ∈ (Fil‘𝑋))
47		isflf 23908	. . . . 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 1373	. . . 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 23152	. . . . 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 800	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 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ∅c0 4280  {csn 4573  ∪ cuni 4856   “ cima 5617  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Topctop 22808  TopOnctopon 22825  neicnei 23012   CnP ccnp 23140  Filcfil 23760   fLim cflim 23849   fLimf cflf 23850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-fbas 21288  df-fg 21289  df-top 22809  df-topon 22826  df-ntr 22935  df-nei 23013  df-cnp 23143  df-fil 23761  df-fm 23853  df-flim 23854  df-flf 23855

This theorem is referenced by:  cnpflf  23916