Theorem restcld 23232

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 22966	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
4		ssexg 5279	. . . . 5 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V)
5	1, 3, 4	syl2anr 606	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
6		resttop 23220	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top)
7	5, 6	syldan 600	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top)
8		eqid 2762	. . . 4 ⊢ ∪ (𝐽 ↾t 𝑆) = ∪ (𝐽 ↾t 𝑆)
9	8	iscld 23087	. . 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 23222	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
12	11	sseq2d 3968	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ ∪ (𝐽 ↾t 𝑆)))
13	11	difeq1d 4079	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝐴) = (∪ (𝐽 ↾t 𝑆) ∖ 𝐴))
14	13	eleq1d 2847	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))
15	12, 14	anbi12d 641	. 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 17456	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
17	5, 16	syldan 600	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
18	17	anbi2d 639	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆))))
19	2	opncld 23093	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
20	19	ad5ant14 767	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
21		incom 4161	. . . . . . . . . . . 12 ⊢ (𝑋 ∩ 𝑆) = (𝑆 ∩ 𝑋)
22		dfss2 3922	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
23	22	biimpi 218	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∩ 𝑋) = 𝑆)
24	21, 23	eqtrid 2809	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∩ 𝑆) = 𝑆)
25	24	ad4antlr 743	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∩ 𝑆) = 𝑆)
26	25	difeq1d 4079	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ((𝑋 ∩ 𝑆) ∖ 𝑜) = (𝑆 ∖ 𝑜))
27		difeq2 4074	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ (𝑜 ∩ 𝑆)))
28		difindi 4244	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆))
29		difid 4329	. . . . . . . . . . . . 13 ⊢ (𝑆 ∖ 𝑆) = ∅
30	29	uneq2i 4118	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ ∅)
31		un0 4348	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ ∅) = (𝑆 ∖ 𝑜)
32	28, 30, 31	3eqtri 2789	. . . . . . . . . . 11 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = (𝑆 ∖ 𝑜)
33	27, 32	eqtrdi 2813	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
34	33	adantl 485	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
35		dfss4 4221	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
36	35	biimpi 218	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑆 → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
37	36	ad3antlr 741	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
38	26, 34, 37	3eqtr2rd 2804	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∩ 𝑆) ∖ 𝑜))
39	21	difeq1i 4076	. . . . . . . . 9 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
40		indif2 4233	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
41		incom 4161	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
42	39, 40, 41	3eqtr2i 2791	. . . . . . . 8 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
43	38, 42	eqtrdi 2813	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆))
44		ineq1 4165	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ 𝑜) ∩ 𝑆))
45	44	rspceeqv 3604	. . . . . . 7 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
46	20, 43, 45	syl2anc 593	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
47	46	rexlimdva2 3165	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) → (∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
48	47	expimpd 457	. . . 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 4244	. . . . . . . . . 10 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆))
51	29	uneq2i 4118	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ ∅)
52		un0 4348	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ ∅) = (𝑆 ∖ 𝑥)
53	50, 51, 52	3eqtri 2789	. . . . . . . . 9 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = (𝑆 ∖ 𝑥)
54		difin2 4253	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
55	54	adantl 485	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
56	53, 55	eqtrid 2809	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
57	56	adantr 484	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
58		simpll 776	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
59	5	adantr 484	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
60	2	cldopn 23091	. . . . . . . . 9 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
61	60	adantl 485	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
62		elrestr 17457	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋 ∖ 𝑥) ∈ 𝐽) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
63	58, 59, 61, 62	syl3anc 1390	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
64	57, 63	eqeltrd 2862	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))
65		inss2 4189	. . . . . 6 ⊢ (𝑥 ∩ 𝑆) ⊆ 𝑆
66	64, 65	jctil 527	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
67		sseq1 3961	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ↔ (𝑥 ∩ 𝑆) ⊆ 𝑆))
68		difeq2 4074	. . . . . . 7 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝑆 ∖ 𝐴) = (𝑆 ∖ (𝑥 ∩ 𝑆)))
69	68	eleq1d 2847	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
70	67, 69	anbi12d 641	. . . . 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 3163	. . 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 399   = wceq 1560   ∈ wcel 2142  ∃wrex 3086  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4865  ‘cfv 6521  (class class class)co 7396   ↾_t crest 17449  Topctop 22953  Clsdccld 23076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-en 8928  df-fin 8931  df-fi 9357  df-rest 17451  df-topgen 17472  df-top 22954  df-topon 22971  df-bases 23006  df-cld 23079

This theorem is referenced by:  restcldi  23233  restcldr  23234  restcls  23241  connsubclo  23484  cldllycmp  23555  iscnrm3rlem2  49562