Theorem fbunfip 23807

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 9442	. . . 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 318	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ¬ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
3		3ioran 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)))
4		df-3an 1088	. . . 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 276	. . 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 289	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
7		nesym 2988	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∅ = (𝑥 ∩ 𝑦))
8	7	ralbii 3082	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦))
9		ralnex 3062	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
10	8, 9	bitri 275	. . . . 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 3062	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
13	11, 12	bitri 275	. . 3 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
14		fbasfip 23806	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
15		fbasfip 23806	. . . . 5 ⊢ (𝐺 ∈ (fBas‘𝑌) → ¬ ∅ ∈ (fi‘𝐺))
16	14, 15	anim12i 613	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)))
17	16	biantrurd 532	. . 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 284	. 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 9431	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (fi‘𝐹))
20		ssralv 4027	. . . . 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 9431	. . . . . 6 ⊢ (𝐺 ∈ (fBas‘𝑌) → 𝐺 ⊆ (fi‘𝐺))
23		ssralv 4027	. . . . . 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 3154	. . . 4 ⊢ (𝐺 ∈ (fBas‘𝑌) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
26	21, 25	sylan9 507	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
27		ineq1 4188	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
28	27	neeq1d 2991	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑦) ≠ ∅))
29		ineq2 4189	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝑤))
30	29	neeq1d 2991	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑤) ≠ ∅))
31	28, 30	cbvral2vw 3224	. . . 4 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅)
32		fbssfi 23775	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥)
33		fbssfi 23775	. . . . . . 7 ⊢ ((𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺)) → ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦)
34		r19.29 3101	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥) → ∃𝑧 ∈ 𝐹 (∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥))
35		r19.29 3101	. . . . . . . . . . . . 13 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → ∃𝑤 ∈ 𝐺 ((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦))
36		ss2in 4220	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → (𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦))
37		sseq2 3985	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ (𝑧 ∩ 𝑤) ⊆ ∅))
38		ss0 4377	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∩ 𝑤) ⊆ ∅ → (𝑧 ∩ 𝑤) = ∅)
39	37, 38	biimtrdi 253	. . . . . . . . . . . . . . . . . . 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 2953	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
42	41	ex 412	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
45	44	rexlimivw 3137	. . . . . . . . . . . . 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 451	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
48	47	rexlimivw 3137	. . . . . . . . . 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 453	. . . . . . . 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 596	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) ∧ (𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
53	52	an4s 660	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) ∧ (𝑥 ∈ (fi‘𝐹) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
54	53	ralrimdvva 3196	. . . 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 242	. . 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 212	. 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 305	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  ‘cfv 6531  ficfi 9422  fBascfbas 21303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1o 8480  df-2o 8481  df-en 8960  df-fin 8963  df-fi 9423  df-fbas 21312

This theorem is referenced by:  isufil2  23846  ufileu  23857  filufint  23858  fmfnfm  23896  hausflim  23919  flimclslem  23922  fclsfnflim  23965  flimfnfcls  23966