Theorem fbunfip 21673

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 8336	. . . 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 308	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ¬ (∅ ∈ (fi‘𝐹) ∨ ∅ ∈ (fi‘𝐺) ∨ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
3		3ioran 1056	. . . 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 1039	. . . 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 265	. . 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	syl6bbr 278	. 2 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ((¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)) ∧ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))))
7		nesym 2850	. . . . . . 7 ⊢ ((𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∅ = (𝑥 ∩ 𝑦))
8	7	ralbii 2980	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦))
9		ralnex 2992	. . . . . 6 ⊢ (∀𝑦 ∈ (fi‘𝐺) ¬ ∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
10	8, 9	bitri 264	. . . . 5 ⊢ (∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
11	10	ralbii 2980	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
12		ralnex 2992	. . . 4 ⊢ (∀𝑥 ∈ (fi‘𝐹) ¬ ∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦) ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
13	11, 12	bitri 264	. . 3 ⊢ (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ ↔ ¬ ∃𝑥 ∈ (fi‘𝐹)∃𝑦 ∈ (fi‘𝐺)∅ = (𝑥 ∩ 𝑦))
14		fbasfip 21672	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))
15		fbasfip 21672	. . . . 5 ⊢ (𝐺 ∈ (fBas‘𝑌) → ¬ ∅ ∈ (fi‘𝐺))
16	14, 15	anim12i 590	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘𝐹) ∧ ¬ ∅ ∈ (fi‘𝐺)))
17	16	biantrurd 529	. . 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	syl5rbb 273	. 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 8325	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (fi‘𝐹))
20		ssralv 3666	. . . . 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 8325	. . . . . 6 ⊢ (𝐺 ∈ (fBas‘𝑌) → 𝐺 ⊆ (fi‘𝐺))
23		ssralv 3666	. . . . . 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 2963	. . . 4 ⊢ (𝐺 ∈ (fBas‘𝑌) → (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
26	21, 25	sylan9 689	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅ → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
27		ineq1 3807	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑧 ∩ 𝑦))
28	27	neeq1d 2853	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑥 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑦) ≠ ∅))
29		ineq2 3808	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑧 ∩ 𝑦) = (𝑧 ∩ 𝑤))
30	29	neeq1d 2853	. . . . 5 ⊢ (𝑦 = 𝑤 → ((𝑧 ∩ 𝑦) ≠ ∅ ↔ (𝑧 ∩ 𝑤) ≠ ∅))
31	28, 30	cbvral2v 3179	. . . 4 ⊢ (∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅ ↔ ∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅)
32		fbssfi 21641	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) → ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥)
33		fbssfi 21641	. . . . . . 7 ⊢ ((𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺)) → ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦)
34		r19.29 3072	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑧 ∈ 𝐹 𝑧 ⊆ 𝑥) → ∃𝑧 ∈ 𝐹 (∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥))
35		r19.29 3072	. . . . . . . . . . . . 13 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ ∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦) → ∃𝑤 ∈ 𝐺 ((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦))
36		ss2in 3840	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → (𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦))
37		sseq2 3627	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∩ 𝑦) = ∅ → ((𝑧 ∩ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ (𝑧 ∩ 𝑤) ⊆ ∅))
38		ss0 3974	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∩ 𝑤) ⊆ ∅ → (𝑧 ∩ 𝑤) = ∅)
39	37, 38	syl6bi 243	. . . . . . . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ⊆ 𝑥 ∧ 𝑤 ⊆ 𝑦) → ((𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
42	41	ex 450	. . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑤 ⊆ 𝑦) → (𝑧 ⊆ 𝑥 → (𝑥 ∩ 𝑦) ≠ ∅))
45	44	rexlimivw 3029	. . . . . . . . . . . . 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 456	. . . . . . . . . . 11 ⊢ ((∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ ∧ 𝑧 ⊆ 𝑥) → (∃𝑤 ∈ 𝐺 𝑤 ⊆ 𝑦 → (𝑥 ∩ 𝑦) ≠ ∅))
48	47	rexlimivw 3029	. . . . . . . . . 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 629	. . . . . . . 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 494	. . . . . 6 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝑥 ∈ (fi‘𝐹)) ∧ (𝐺 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
53	52	an4s 869	. . . . 5 ⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) ∧ (𝑥 ∈ (fi‘𝐹) ∧ 𝑦 ∈ (fi‘𝐺))) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → (𝑥 ∩ 𝑦) ≠ ∅))
54	53	ralrimdvva 2974	. . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (∀𝑧 ∈ 𝐹 ∀𝑤 ∈ 𝐺 (𝑧 ∩ 𝑤) ≠ ∅ → ∀𝑥 ∈ (fi‘𝐹)∀𝑦 ∈ (fi‘𝐺)(𝑥 ∩ 𝑦) ≠ ∅))
55	31, 54	syl5bi 232	. . 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 202	. 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 294	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ‘cfv 5888  ficfi 8316  fBascfbas 19734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743

This theorem is referenced by:  isufil2  21712  ufileu  21723  filufint  21724  fmfnfm  21762  hausflim  21785  flimclslem  21788  fclsfnflim  21831  flimfnfcls  21832