Theorem restcld 23297

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 23	. . . . 5 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 ⊆ 𝑋)
2		restcld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	topopn 23031	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
4		ssexg 5294	. . . . 5 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V)
5	1, 3, 4	syl2anr 608	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
6		resttop 23285	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top)
7	5, 6	syldan 602	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top)
8		eqid 2769	. . . 4 ⊢ ∪ (𝐽 ↾t 𝑆) = ∪ (𝐽 ↾t 𝑆)
9	8	iscld 23152	. . 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 18	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ ∪ (𝐽 ↾t 𝑆) ∧ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
11	2	restuni 23287	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
12	11	sseq2d 3977	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ ∪ (𝐽 ↾t 𝑆)))
13	11	difeq1d 4088	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝐴) = (∪ (𝐽 ↾t 𝑆) ∖ 𝐴))
14	13	eleq1d 2854	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))
15	12, 14	anbi12d 643	. 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 17479	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
17	5, 16	syldan 602	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
18	17	anbi2d 641	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆))))
19	2	opncld 23158	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
20	19	ad5ant14 769	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
21		incom 4170	. . . . . . . . . . . 12 ⊢ (𝑋 ∩ 𝑆) = (𝑆 ∩ 𝑋)
22		dfss2 3931	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
23	22	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∩ 𝑋) = 𝑆)
24	21, 23	eqtrid 2816	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∩ 𝑆) = 𝑆)
25	24	ad4antlr 745	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∩ 𝑆) = 𝑆)
26	25	difeq1d 4088	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ((𝑋 ∩ 𝑆) ∖ 𝑜) = (𝑆 ∖ 𝑜))
27		difeq2 4083	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ (𝑜 ∩ 𝑆)))
28		difindi 4253	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆))
29		difid 4339	. . . . . . . . . . . . 13 ⊢ (𝑆 ∖ 𝑆) = ∅
30	29	uneq2i 4127	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ ∅)
31		un0 4358	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ ∅) = (𝑆 ∖ 𝑜)
32	28, 30, 31	3eqtri 2796	. . . . . . . . . . 11 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = (𝑆 ∖ 𝑜)
33	27, 32	eqtrdi 2820	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
34	33	adantl 486	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
35		dfss4 4230	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
36	35	biimpi 219	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑆 → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
37	36	ad3antlr 743	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
38	26, 34, 37	3eqtr2rd 2811	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∩ 𝑆) ∖ 𝑜))
39	21	difeq1i 4085	. . . . . . . . 9 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
40		indif2 4242	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
41		incom 4170	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
42	39, 40, 41	3eqtr2i 2798	. . . . . . . 8 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
43	38, 42	eqtrdi 2820	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆))
44		ineq1 4174	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ 𝑜) ∩ 𝑆))
45	44	rspceeqv 3613	. . . . . . 7 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
46	20, 43, 45	syl2anc 595	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
47	46	rexlimdva2 3174	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) → (∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
48	47	expimpd 458	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
49	18, 48	sylbid 243	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
50		difindi 4253	. . . . . . . . . 10 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆))
51	29	uneq2i 4127	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ ∅)
52		un0 4358	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ ∅) = (𝑆 ∖ 𝑥)
53	50, 51, 52	3eqtri 2796	. . . . . . . . 9 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = (𝑆 ∖ 𝑥)
54		difin2 4262	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
55	54	adantl 486	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
56	53, 55	eqtrid 2816	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
57	56	adantr 485	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
58		simpll 778	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
59	5	adantr 485	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
60	2	cldopn 23156	. . . . . . . . 9 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
61	60	adantl 486	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
62		elrestr 17480	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋 ∖ 𝑥) ∈ 𝐽) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
63	58, 59, 61, 62	syl3anc 1396	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
64	57, 63	eqeltrd 2869	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))
65		inss2 4198	. . . . . 6 ⊢ (𝑥 ∩ 𝑆) ⊆ 𝑆
66	64, 65	jctil 528	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
67		sseq1 3970	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ↔ (𝑥 ∩ 𝑆) ⊆ 𝑆))
68		difeq2 4083	. . . . . . 7 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝑆 ∖ 𝐴) = (𝑆 ∖ (𝑥 ∩ 𝑆)))
69	68	eleq1d 2854	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
70	67, 69	anbi12d 643	. . . . 5 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))))
71	66, 70	syl5ibrcom 250	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
72	71	rexlimdva 3172	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
73	49, 72	impbid 215	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
74	10, 15, 73	3bitr2d 310	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4876  ‘cfv 6537  (class class class)co 7411   ↾_t crest 17472  Topctop 23018  Clsdccld 23141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-en 8943  df-fin 8946  df-fi 9370  df-rest 17474  df-topgen 17495  df-top 23019  df-topon 23036  df-bases 23071  df-cld 23144

This theorem is referenced by:  restcldi  23298  restcldr  23299  restcls  23306  connsubclo  23549  cldllycmp  23620  iscnrm3rlem2  49603