Theorem restcld 21782

Description: A closed set of a subspace topology is a closed set of the original topology intersected with the subset. (Contributed by FL, 11-Jul-2009.) (Proof shortened by Mario Carneiro, 15-Dec-2013.)

Hypothesis

Ref	Expression
restcld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
restcld	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem restcld

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 ⊆ 𝑋)
2		restcld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	topopn 21516	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
4		ssexg 5229	. . . . 5 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V)
5	1, 3, 4	syl2anr 598	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
6		resttop 21770	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top)
7	5, 6	syldan 593	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top)
8		eqid 2823	. . . 4 ⊢ ∪ (𝐽 ↾t 𝑆) = ∪ (𝐽 ↾t 𝑆)
9	8	iscld 21637	. . 3 ⊢ ((𝐽 ↾t 𝑆) ∈ Top → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
10	7, 9	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
11	2	restuni 21772	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
12	11	sseq2d 4001	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ ∪ (𝐽 ↾t 𝑆)))
13	11	difeq1d 4100	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝐴) = (∪ (𝐽 ↾t 𝑆) ∖ 𝐴))
14	13	eleq1d 2899	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))
15	12, 14	anbi12d 632	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
16		elrest 16703	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
17	5, 16	syldan 593	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
18	17	anbi2d 630	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆))))
19	2	opncld 21643	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
20	19	ad5ant14 756	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
21		incom 4180	. . . . . . . . . . . 12 ⊢ (𝑋 ∩ 𝑆) = (𝑆 ∩ 𝑋)
22		df-ss 3954	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
23	22	biimpi 218	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∩ 𝑋) = 𝑆)
24	21, 23	syl5eq 2870	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∩ 𝑆) = 𝑆)
25	24	ad4antlr 731	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∩ 𝑆) = 𝑆)
26	25	difeq1d 4100	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ((𝑋 ∩ 𝑆) ∖ 𝑜) = (𝑆 ∖ 𝑜))
27		difeq2 4095	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ (𝑜 ∩ 𝑆)))
28		difindi 4260	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆))
29		difid 4332	. . . . . . . . . . . . 13 ⊢ (𝑆 ∖ 𝑆) = ∅
30	29	uneq2i 4138	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ ∅)
31		un0 4346	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ ∅) = (𝑆 ∖ 𝑜)
32	28, 30, 31	3eqtri 2850	. . . . . . . . . . 11 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = (𝑆 ∖ 𝑜)
33	27, 32	syl6eq 2874	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
34	33	adantl 484	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
35		dfss4 4237	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
36	35	biimpi 218	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑆 → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
37	36	ad3antlr 729	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
38	26, 34, 37	3eqtr2rd 2865	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∩ 𝑆) ∖ 𝑜))
39	21	difeq1i 4097	. . . . . . . . 9 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
40		indif2 4249	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
41		incom 4180	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
42	39, 40, 41	3eqtr2i 2852	. . . . . . . 8 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
43	38, 42	syl6eq 2874	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆))
44		ineq1 4183	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ 𝑜) ∩ 𝑆))
45	44	rspceeqv 3640	. . . . . . 7 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
46	20, 43, 45	syl2anc 586	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
47	46	rexlimdva2 3289	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) → (∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
48	47	expimpd 456	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
49	18, 48	sylbid 242	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
50		difindi 4260	. . . . . . . . . 10 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆))
51	29	uneq2i 4138	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ ∅)
52		un0 4346	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ ∅) = (𝑆 ∖ 𝑥)
53	50, 51, 52	3eqtri 2850	. . . . . . . . 9 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = (𝑆 ∖ 𝑥)
54		difin2 4268	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
55	54	adantl 484	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
56	53, 55	syl5eq 2870	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
57	56	adantr 483	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
58		simpll 765	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
59	5	adantr 483	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
60	2	cldopn 21641	. . . . . . . . 9 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
61	60	adantl 484	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
62		elrestr 16704	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋 ∖ 𝑥) ∈ 𝐽) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
63	58, 59, 61, 62	syl3anc 1367	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
64	57, 63	eqeltrd 2915	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))
65		inss2 4208	. . . . . 6 ⊢ (𝑥 ∩ 𝑆) ⊆ 𝑆
66	64, 65	jctil 522	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
67		sseq1 3994	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ↔ (𝑥 ∩ 𝑆) ⊆ 𝑆))
68		difeq2 4095	. . . . . . 7 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝑆 ∖ 𝐴) = (𝑆 ∖ (𝑥 ∩ 𝑆)))
69	68	eleq1d 2899	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
70	67, 69	anbi12d 632	. . . . 5 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))))
71	66, 70	syl5ibrcom 249	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
72	71	rexlimdva 3286	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
73	49, 72	impbid 214	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
74	10, 15, 73	3bitr2d 309	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  ∪ cuni 4840  ‘cfv 6357  (class class class)co 7158   ↾_t crest 16696  Topctop 21503  Clsdccld 21626

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-oadd 8108  df-er 8291  df-en 8512  df-fin 8515  df-fi 8877  df-rest 16698  df-topgen 16719  df-top 21504  df-topon 21521  df-bases 21556  df-cld 21629

This theorem is referenced by:  restcldi  21783  restcldr  21784  restcls  21791  connsubclo  22034  cldllycmp  22105