Theorem restcld 23398

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 23132	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
4		ssexg 5284	. . . . 5 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑆 ∈ V)
5	1, 3, 4	syl2anr 609	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
6		resttop 23386	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝐽 ↾t 𝑆) ∈ Top)
7	5, 6	syldan 603	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐽 ↾t 𝑆) ∈ Top)
8		eqid 2760	. . . 4 ⊢ ∪ (𝐽 ↾t 𝑆) = ∪ (𝐽 ↾t 𝑆)
9	8	iscld 23253	. . 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 23388	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾t 𝑆))
12	11	sseq2d 3963	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝐴 ⊆ 𝑆 ↔ 𝐴 ⊆ ∪ (𝐽 ↾t 𝑆)))
13	11	difeq1d 4073	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝐴) = (∪ (𝐽 ↾t 𝑆) ∖ 𝐴))
14	13	eleq1d 2845	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (∪ (𝐽 ↾t 𝑆) ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)))
15	12, 14	anbi12d 644	. 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 17513	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
17	5, 16	syldan 603	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)))
18	17	anbi2d 642	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐴 ⊆ 𝑆 ∧ (𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆)) ↔ (𝐴 ⊆ 𝑆 ∧ ∃𝑜 ∈ 𝐽 (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆))))
19	2	opncld 23259	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
20	19	ad5ant14 770	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
21		incom 4155	. . . . . . . . . . . 12 ⊢ (𝑋 ∩ 𝑆) = (𝑆 ∩ 𝑋)
22		dfss2 3917	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑆 ∩ 𝑋) = 𝑆)
23	22	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∩ 𝑋) = 𝑆)
24	21, 23	eqtrid 2807	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∩ 𝑆) = 𝑆)
25	24	ad4antlr 746	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑋 ∩ 𝑆) = 𝑆)
26	25	difeq1d 4073	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → ((𝑋 ∩ 𝑆) ∖ 𝑜) = (𝑆 ∖ 𝑜))
27		difeq2 4068	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ (𝑜 ∩ 𝑆)))
28		difindi 4238	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆))
29		difid 4325	. . . . . . . . . . . . 13 ⊢ (𝑆 ∖ 𝑆) = ∅
30	29	uneq2i 4112	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑜) ∪ ∅)
31		un0 4344	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ 𝑜) ∪ ∅) = (𝑆 ∖ 𝑜)
32	28, 30, 31	3eqtri 2787	. . . . . . . . . . 11 ⊢ (𝑆 ∖ (𝑜 ∩ 𝑆)) = (𝑆 ∖ 𝑜)
33	27, 32	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
34	33	adantl 487	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = (𝑆 ∖ 𝑜))
35		dfss4 4215	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
36	35	biimpi 219	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑆 → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
37	36	ad3antlr 744	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → (𝑆 ∖ (𝑆 ∖ 𝐴)) = 𝐴)
38	26, 34, 37	3eqtr2rd 2802	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∩ 𝑆) ∖ 𝑜))
39	21	difeq1i 4070	. . . . . . . . 9 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
40		indif2 4227	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑆 ∩ 𝑋) ∖ 𝑜)
41		incom 4155	. . . . . . . . 9 ⊢ (𝑆 ∩ (𝑋 ∖ 𝑜)) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
42	39, 40, 41	3eqtr2i 2789	. . . . . . . 8 ⊢ ((𝑋 ∩ 𝑆) ∖ 𝑜) = ((𝑋 ∖ 𝑜) ∩ 𝑆)
43	38, 42	eqtrdi 2811	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ⊆ 𝑆) ∧ 𝑜 ∈ 𝐽) ∧ (𝑆 ∖ 𝐴) = (𝑜 ∩ 𝑆)) → 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆))
44		ineq1 4159	. . . . . . . 8 ⊢ (𝑥 = (𝑋 ∖ 𝑜) → (𝑥 ∩ 𝑆) = ((𝑋 ∖ 𝑜) ∩ 𝑆))
45	44	rspceeqv 3599	. . . . . . 7 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((𝑋 ∖ 𝑜) ∩ 𝑆)) → ∃𝑥 ∈ (Clsd‘𝐽)𝐴 = (𝑥 ∩ 𝑆))
46	20, 43, 45	syl2anc 596	. . . . . 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 459	. . . 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 4238	. . . . . . . . . 10 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆))
51	29	uneq2i 4112	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ (𝑆 ∖ 𝑆)) = ((𝑆 ∖ 𝑥) ∪ ∅)
52		un0 4344	. . . . . . . . . 10 ⊢ ((𝑆 ∖ 𝑥) ∪ ∅) = (𝑆 ∖ 𝑥)
53	50, 51, 52	3eqtri 2787	. . . . . . . . 9 ⊢ (𝑆 ∖ (𝑥 ∩ 𝑆)) = (𝑆 ∖ 𝑥)
54		difin2 4247	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
55	54	adantl 487	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ 𝑥) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
56	53, 55	eqtrid 2807	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
57	56	adantr 486	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) = ((𝑋 ∖ 𝑥) ∩ 𝑆))
58		simpll 779	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
59	5	adantr 486	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑆 ∈ V)
60	2	cldopn 23257	. . . . . . . . 9 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
61	60	adantl 487	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
62		elrestr 17514	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V ∧ (𝑋 ∖ 𝑥) ∈ 𝐽) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
63	58, 59, 61, 62	syl3anc 1398	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑋 ∖ 𝑥) ∩ 𝑆) ∈ (𝐽 ↾t 𝑆))
64	57, 63	eqeltrd 2860	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆))
65		inss2 4183	. . . . . 6 ⊢ (𝑥 ∩ 𝑆) ⊆ 𝑆
66	64, 65	jctil 529	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (Clsd‘𝐽)) → ((𝑥 ∩ 𝑆) ⊆ 𝑆 ∧ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
67		sseq1 3956	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝐴 ⊆ 𝑆 ↔ (𝑥 ∩ 𝑆) ⊆ 𝑆))
68		difeq2 4068	. . . . . . 7 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → (𝑆 ∖ 𝐴) = (𝑆 ∖ (𝑥 ∩ 𝑆)))
69	68	eleq1d 2845	. . . . . 6 ⊢ (𝐴 = (𝑥 ∩ 𝑆) → ((𝑆 ∖ 𝐴) ∈ (𝐽 ↾t 𝑆) ↔ (𝑆 ∖ (𝑥 ∩ 𝑆)) ∈ (𝐽 ↾t 𝑆)))
70	67, 69	anbi12d 644	. . . . 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 3163	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ‘cfv 6533  (class class class)co 7414   ↾_t crest 17506  Topctop 23119  Clsdccld 23242

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-en 8954  df-fin 8957  df-fi 9382  df-rest 17508  df-topgen 17529  df-top 23120  df-topon 23137  df-bases 23172  df-cld 23245

This theorem is used by:  restcldi  23399  restcldr  23400  restcls  23407  connsubclo  23650  cldllycmp  23722  iscnrm3rlem2  49868