Theorem restcls 21478

Description: A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)

Hypotheses

Ref	Expression
restcls.1	⊢ 𝑋 = ∪ 𝐽
restcls.2	⊢ 𝐾 = (𝐽 ↾t 𝑌)

Assertion

Ref	Expression
restcls	⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restcls

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1129	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
2		sstr 3901	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
3	2	ancoms 459	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
4	3	3adant1 1123	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
5		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6	5	clscld 21344	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
7	1, 4, 6	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
8		eqid 2795	. . . . 5 ⊢ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9		ineq1 4105	. . . . . 6 ⊢ (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥 ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
10	9	rspceeqv 3577	. . . . 5 ⊢ ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌))
11	7, 8, 10	sylancl 586	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌))
12		restcls.2	. . . . . . 7 ⊢ 𝐾 = (𝐽 ↾t 𝑌)
13	12	fveq2i 6546	. . . . . 6 ⊢ (Clsd‘𝐾) = (Clsd‘(𝐽 ↾t 𝑌))
14	13	eleq2i 2874	. . . . 5 ⊢ ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
15	5	restcld 21469	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌)))
16	15	3adant3 1125	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌)))
17	14, 16	syl5bb 284	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌)))
18	11, 17	mpbird 258	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
19	5	sscls 21353	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
20	1, 4, 19	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
21		simp3 1131	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌)
22	20, 21	ssind 4133	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
23		eqid 2795	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
24	23	clsss2 21369	. . 3 ⊢ (((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
25	18, 22, 24	syl2anc 584	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
26	12	fveq2i 6546	. . . . . 6 ⊢ (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌))
27	26	fveq1i 6544	. . . . 5 ⊢ ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆)
28		id 22	. . . . . . . . 9 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
29	5	topopn 21203	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
30		ssexg 5123	. . . . . . . . 9 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
31	28, 29, 30	syl2anr 596	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
32		resttop 21457	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Top)
33	31, 32	syldan 591	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ Top)
34	33	3adant3 1125	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top)
35	5	restuni 21459	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
36	35	3adant3 1125	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
37	21, 36	sseqtrd 3932	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
38		eqid 2795	. . . . . . 7 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ (𝐽 ↾t 𝑌)
39	38	clscld 21344	. . . . . 6 ⊢ (((𝐽 ↾t 𝑌) ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
40	34, 37, 39	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
41	27, 40	syl5eqel 2887	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
42	5	restcld 21469	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌)))
43	42	3adant3 1125	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌)))
44	41, 43	mpbid 233	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))
45	12, 33	syl5eqel 2887	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
46	45	3adant3 1125	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
47	12	unieqi 4758	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)
48	47	eqcomi 2804	. . . . . . . 8 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ 𝐾
49	48	sscls 21353	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
50	46, 37, 49	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
51	50	adantr 481	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
52		inss1 4129	. . . . . . 7 ⊢ (𝑥 ∩ 𝑌) ⊆ 𝑥
53		sseq1 3917	. . . . . . 7 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ 𝑌) ⊆ 𝑥))
54	52, 53	mpbiri 259	. . . . . 6 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
55	54	ad2antll 725	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
56	51, 55	sstrd 3903	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ 𝑥)
57	5	clsss2 21369	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
58	57	adantl 482	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
59	58	ssrind 4136	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌))
60		sseq2 3918	. . . . . . . 8 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌)))
61	59, 60	syl5ibrcom 248	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
62	61	expr 457	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
63	62	com23 86	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (𝑆 ⊆ 𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
64	63	impr 455	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → (𝑆 ⊆ 𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
65	56, 64	mpd 15	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
66	44, 65	rexlimddv 3254	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
67	25, 66	eqssd 3910	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∃wrex 3106  Vcvv 3437   ∩ cin 3862   ⊆ wss 3863  ∪ cuni 4749  ‘cfv 6230  (class class class)co 7021   ↾_t crest 16528  Topctop 21190  Clsdccld 21313  clsccl 21315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-iin 4832  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-oadd 7962  df-er 8144  df-en 8363  df-fin 8366  df-fi 8726  df-rest 16530  df-topgen 16551  df-top 21191  df-topon 21208  df-bases 21243  df-cld 21316  df-cls 21318

This theorem is referenced by:  restlp  21480  resscdrg  23649