Theorem restcld 23059

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 22793	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
4		ssexg 5278	. . . . 5 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V)
5	1, 3, 4	syl2anr 597	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
6		resttop 23047	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top)
7	5, 6	syldan 591	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top)
8		eqid 2729	. . . 4 ⊢ ∪ (𝐽 ↾t 𝑆) = ∪ (𝐽 ↾t 𝑆)
9	8	iscld 22914	. . 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 23049	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
12	11	sseq2d 3979	. . 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 2813	. . 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 17390	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
17	5, 16	syldan 591	. . . . 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 22920	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
20	19	ad5ant14 757	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
21		incom 4172	. . . . . . . . . . . 12 ⊢ (𝑋 ∩ 𝑆) = (𝑆 ∩ 𝑋)
22		dfss2 3932	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
23	22	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∩ 𝑋) = 𝑆)
24	21, 23	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∩ 𝑆) = 𝑆)
25	24	ad4antlr 733	. . . . . . . . . 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 4255	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆))
29		difid 4339	. . . . . . . . . . . . 13 ⊢ (𝑆 ∖ 𝑆) = ∅
30	29	uneq2i 4128	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ ∅)
31		un0 4357	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ ∅) = (𝑆 ∖ 𝑜)
32	28, 30, 31	3eqtri 2756	. . . . . . . . . . 11 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = (𝑆 ∖ 𝑜)
33	27, 32	eqtrdi 2780	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
34	33	adantl 481	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
35		dfss4 4232	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
36	35	biimpi 216	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑆 → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
37	36	ad3antlr 731	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
38	26, 34, 37	3eqtr2rd 2771	. . . . . . . 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 4244	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
41		incom 4172	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
42	39, 40, 41	3eqtr2i 2758	. . . . . . . 8 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
43	38, 42	eqtrdi 2780	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆))
44		ineq1 4176	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ 𝑜) ∩ 𝑆))
45	44	rspceeqv 3611	. . . . . . 7 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
46	20, 43, 45	syl2anc 584	. . . . . 6 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
47	46	rexlimdva2 3136	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) → (∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
48	47	expimpd 453	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
49	18, 48	sylbid 240	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
50		difindi 4255	. . . . . . . . . 10 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆))
51	29	uneq2i 4128	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ ∅)
52		un0 4357	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ ∅) = (𝑆 ∖ 𝑥)
53	50, 51, 52	3eqtri 2756	. . . . . . . . 9 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = (𝑆 ∖ 𝑥)
54		difin2 4264	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
55	54	adantl 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
56	53, 55	eqtrid 2776	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
57	56	adantr 480	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
58		simpll 766	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
59	5	adantr 480	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
60	2	cldopn 22918	. . . . . . . . 9 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
61	60	adantl 481	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
62		elrestr 17391	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋 ∖ 𝑥) ∈ 𝐽) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
63	58, 59, 61, 62	syl3anc 1373	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
64	57, 63	eqeltrd 2828	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))
65		inss2 4201	. . . . . 6 ⊢ (𝑥 ∩ 𝑆) ⊆ 𝑆
66	64, 65	jctil 519	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
67		sseq1 3972	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ↔ (𝑥 ∩ 𝑆) ⊆ 𝑆))
68		difeq2 4083	. . . . . . 7 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝑆 ∖ 𝐴) = (𝑆 ∖ (𝑥 ∩ 𝑆)))
69	68	eleq1d 2813	. . . . . 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 247	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
72	71	rexlimdva 3134	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆))))
73	49, 72	impbid 212	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))
74	10, 15, 73	3bitr2d 307	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘(𝐽 ↾t 𝑆)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  Clsdccld 22903

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-en 8919  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cld 22906

This theorem is referenced by:  restcldi  23060  restcldr  23061  restcls  23068  connsubclo  23311  cldllycmp  23382  iscnrm3rlem2  48929