Theorem restcls 23410

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 1154	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
2		sstr 3942	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
3	2	ancoms 464	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
4	3	3adant1 1148	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
5		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6	5	clscld 23276	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
7	1, 4, 6	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
8		eqid 2762	. . . . 5 ⊢ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9		ineq1 4162	. . . . . 6 ⊢ (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥 ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
10	9	rspceeqv 3602	. . . . 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 598	. . . 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 6885	. . . . . 6 ⊢ (Clsd‘𝐾) = (Clsd‘(𝐽 ↾t 𝑌))
14	13	eleq2i 2854	. . . . 5 ⊢ ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
15	5	restcld 23401	. . . . . 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 1150	. . . . 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	bitrid 286	. . . 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 260	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
19	5	sscls 23285	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
20	1, 4, 19	syl2anc 596	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
21		simp3 1156	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌)
22	20, 21	ssind 4189	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
23		eqid 2762	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
24	23	clsss2 23301	. . 3 ⊢ (((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
25	18, 22, 24	syl2anc 596	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
26	12	fveq2i 6885	. . . . . 6 ⊢ (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌))
27	26	fveq1i 6883	. . . . 5 ⊢ ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆)
28		id 23	. . . . . . . . 9 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
29	5	topopn 23135	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
30		ssexg 5288	. . . . . . . . 9 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
31	28, 29, 30	syl2anr 609	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
32		resttop 23389	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Top)
33	31, 32	syldan 603	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ Top)
34	33	3adant3 1150	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top)
35	5	restuni 23391	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
36	35	3adant3 1150	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
37	21, 36	sseqtrd 3970	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
38		eqid 2762	. . . . . . 7 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ (𝐽 ↾t 𝑌)
39	38	clscld 23276	. . . . . 6 ⊢ (((𝐽 ↾t 𝑌) ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
40	34, 37, 39	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
41	27, 40	eqeltrid 2866	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
42	5	restcld 23401	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌)))
43	42	3adant3 1150	. . . 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 235	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))
45	12, 33	eqeltrid 2866	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
46	45	3adant3 1150	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
47	12	unieqi 4882	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)
48	47	eqcomi 2771	. . . . . . . 8 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ 𝐾
49	48	sscls 23285	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
50	46, 37, 49	syl2anc 596	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
51	50	adantr 486	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
52		inss1 4185	. . . . . . 7 ⊢ (𝑥 ∩ 𝑌) ⊆ 𝑥
53		sseq1 3959	. . . . . . 7 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ 𝑌) ⊆ 𝑥))
54	52, 53	mpbiri 261	. . . . . 6 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
55	54	ad2antll 742	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
56	51, 55	sstrd 3944	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ 𝑥)
57	5	clsss2 23301	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
58	57	adantl 487	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
59	58	ssrind 4192	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌))
60		sseq2 3960	. . . . . . . 8 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌)))
61	59, 60	syl5ibrcom 250	. . . . . . 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 462	. . . . . 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 87	. . . . 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 460	. . . 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 16	. . 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 3171	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
67	25, 66	eqssd 3951	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∪ cuni 4870  ‘cfv 6537  (class class class)co 7416   ↾_t crest 17509  Topctop 23122  Clsdccld 23245  clsccl 23247

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-en 8956  df-fin 8959  df-fi 9384  df-rest 17511  df-topgen 17532  df-top 23123  df-topon 23140  df-bases 23175  df-cld 23248  df-cls 23250

This theorem is used by:  restlp  23412  resscdrg  25590  restcls2  49842