Theorem fbunfip 23817

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 9455	. . . 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 317	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ¬ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
3		3ioran 1103	. . . 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 1086	. . . 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 275	. . 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 288	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
7		nesym 2986	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∅ = (𝑥 ∩ 𝑦))
8	7	ralbii 3082	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦))
9		ralnex 3061	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
10	8, 9	bitri 274	. . . . 5 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
11	10	ralbii 3082	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
12		ralnex 3061	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
13	11, 12	bitri 274	. . 3 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
14		fbasfip 23816	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
15		fbasfip 23816	. . . . 5 ⊢ (𝐺 ∈ (fBas‘𝑌) → ¬ ∅ ∈ (fi‘𝐺))
16	14, 15	anim12i 611	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)))
17	16	biantrurd 531	. . 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 283	. 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 9444	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (fi‘𝐹))
20		ssralv 4045	. . . . 5 ⊢ (𝐹 ⊆ (fi‘𝐹) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
21	19, 20	syl 17	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
22		ssfii 9444	. . . . . 6 ⊢ (𝐺 ∈ (fBas‘𝑌) → 𝐺 ⊆ (fi‘𝐺))
23		ssralv 4045	. . . . . 6 ⊢ (𝐺 ⊆ (fi‘𝐺) → (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
24	22, 23	syl 17	. . . . 5 ⊢ (𝐺 ∈ (fBas‘𝑌) → (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
25	24	ralimdv 3158	. . . 4 ⊢ (𝐺 ∈ (fBas‘𝑌) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
26	21, 25	sylan9 506	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
27		ineq1 4203	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
28	27	neeq1d 2989	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑦) ≠ ∅))
29		ineq2 4204	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝑤))
30	29	neeq1d 2989	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑤) ≠ ∅))
31	28, 30	cbvral2vw 3228	. . . 4 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅)
32		fbssfi 23785	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥)
33		fbssfi 23785	. . . . . . 7 ⊢ ((𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺)) → ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦)
34		r19.29 3103	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥) → ∃𝑧 ∈ 𝐹 (∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥))
35		r19.29 3103	. . . . . . . . . . . . 13 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → ∃𝑤 ∈ 𝐺 ((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦))
36		ss2in 4235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → (𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦))
37		sseq2 4003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ (𝑧 ∩ 𝑤) ⊆ ∅))
38		ss0 4400	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∩ 𝑤) ⊆ ∅ → (𝑧 ∩ 𝑤) = ∅)
39	37, 38	biimtrdi 252	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦) → (𝑧 ∩ 𝑤) = ∅))
40	36, 39	syl5com 31	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑧 ∩ 𝑤) = ∅))
41	40	necon3d 2950	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
42	41	ex 411	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ 𝑥 → (𝑤 ⊆ 𝑦 → ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅)))
43	42	com13 88	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑤 ⊆ 𝑦 → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅)))
44	43	imp 405	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
45	44	rexlimivw 3140	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ 𝐺 ((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
46	35, 45	syl 17	. . . . . . . . . . . 12 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
47	46	impancom 450	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
48	47	rexlimivw 3140	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐹 (∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
49	34, 48	syl 17	. . . . . . . . 9 ⊢ ((∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
50	49	expimpd 452	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥 ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → (𝑥 ∩ 𝑦) ≠ ∅))
51	50	com12 32	. . . . . . 7 ⊢ ((∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥 ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
52	32, 33, 51	syl2an 594	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) ∧ (𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
53	52	an4s 658	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) ∧ (𝑥 ∈ (fi‘𝐹) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
54	53	ralrimdvva 3199	. . . 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 241	. . 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 211	. 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 304	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4322  ‘cfv 6549  ficfi 9435  fBascfbas 21284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-er 8725  df-en 8965  df-fin 8968  df-fi 9436  df-fbas 21293

This theorem is referenced by:  isufil2  23856  ufileu  23867  filufint  23868  fmfnfm  23906  hausflim  23929  flimclslem  23932  fclsfnflim  23975  flimfnfcls  23976