Theorem fbunfip 24026

Description: A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fbunfip	⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Proof of Theorem fbunfip

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfiun 9386	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
2	1	notbid 321	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ¬ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
3		3ioran 1123	. . . 4 ⊢ (¬ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)) ↔ (¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)))
4		df-3an 1105	. . . 4 ⊢ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)) ↔ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)))
5	3, 4	bitr2i 279	. . 3 ⊢ (((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)) ↔ ¬ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)))
6	2, 5	bitr4di 292	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
7		nesym 3014	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∅ = (𝑥 ∩ 𝑦))
8	7	ralbii 3111	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦))
9		ralnex 3091	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
10	8, 9	bitri 278	. . . . 5 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
11	10	ralbii 3111	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
12		ralnex 3091	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
13	11, 12	bitri 278	. . 3 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
14		fbasfip 24025	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
15		fbasfip 24025	. . . . 5 ⊢ (𝐺 ∈ (fBas‘𝑌) → ¬ ∅ ∈ (fi‘𝐺))
16	14, 15	anim12i 624	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)))
17	16	biantrurd 541	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦) ↔ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
18	13, 17	bitr2id 287	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
19		ssfii 9375	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (fi‘𝐹))
20		ssralv 4006	. . . . 5 ⊢ (𝐹 ⊆ (fi‘𝐹) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
21	19, 20	syl 18	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
22		ssfii 9375	. . . . . 6 ⊢ (𝐺 ∈ (fBas‘𝑌) → 𝐺 ⊆ (fi‘𝐺))
23		ssralv 4006	. . . . . 6 ⊢ (𝐺 ⊆ (fi‘𝐺) → (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
24	22, 23	syl 18	. . . . 5 ⊢ (𝐺 ∈ (fBas‘𝑌) → (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
25	24	ralimdv 3179	. . . 4 ⊢ (𝐺 ∈ (fBas‘𝑌) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
26	21, 25	sylan9 516	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
27		ineq1 4166	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
28	27	neeq1d 3017	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑦) ≠ ∅))
29		ineq2 4167	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝑤))
30	29	neeq1d 3017	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑤) ≠ ∅))
31	28, 30	cbvral2vw 3247	. . . 4 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅)
32		fbssfi 23994	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥)
33		fbssfi 23994	. . . . . . 7 ⊢ ((𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺)) → ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦)
34		r19.29 3128	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥) → ∃𝑧 ∈ 𝐹 (∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥))
35		r19.29 3128	. . . . . . . . . . . . 13 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → ∃𝑤 ∈ 𝐺 ((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦))
36		ss2in 4197	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → (𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦))
37		sseq2 3963	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ (𝑧 ∩ 𝑤) ⊆ ∅))
38		ss0 4359	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∩ 𝑤) ⊆ ∅ → (𝑧 ∩ 𝑤) = ∅)
39	37, 38	biimtrdi 256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦) → (𝑧 ∩ 𝑤) = ∅))
40	36, 39	syl5com 32	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑤) = ∅))
41	40	necon3d 2979	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
42	41	ex 417	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝑥 → (𝑤 ⊆ 𝑦 → ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅)))
43	42	com13 89	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑤 ⊆ 𝑦 → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅)))
44	43	imp 411	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
45	44	rexlimivw 3162	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ 𝐺 ((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
46	35, 45	syl 18	. . . . . . . . . . . 12 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
47	46	impancom 456	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
48	47	rexlimivw 3162	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐹 (∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
49	34, 48	syl 18	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
50	49	expimpd 458	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥 ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ≠ ∅))
51	50	com12 33	. . . . . . 7 ⊢ ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥 ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
52	32, 33, 51	syl2an 607	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) ∧ (𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
53	52	an4s 672	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) ∧ (𝑥 ∈ (fi‘𝐹) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
54	53	ralrimdvva 3220	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → ∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
55	31, 54	biimtrid 245	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
56	26, 55	impbid 215	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
57	6, 18, 56	3bitrd 308	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ‘cfv 6536  ficfi 9366  fBascfbas 21510

This theorem was proved from 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-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  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-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  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-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  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-om 7859  df-1o 8449  df-2o 8450  df-en 8940  df-fin 8943  df-fi 9367  df-fbas 21519

This theorem is referenced by:  isufil2  24065  ufileu  24076  filufint  24077  fmfnfm  24115  hausflim  24138  flimclslem  24141  fclsfnflim  24184  flimfnfcls  24185