Theorem restcls 23068

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 1136	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top)
2		sstr 3955	. . . . . . . 8 ⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋)
3	2	ancoms 458	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
4	3	3adant1 1130	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋)
5		restcls.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6	5	clscld 22934	. . . . . 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 2729	. . . . 5 ⊢ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9		ineq1 4176	. . . . . 6 ⊢ (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥 ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
10	9	rspceeqv 3611	. . . . 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 6861	. . . . . 6 ⊢ (Clsd‘𝐾) = (Clsd‘(𝐽 ↾t 𝑌))
14	13	eleq2i 2820	. . . . 5 ⊢ ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
15	5	restcld 23059	. . . . . 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 1132	. . . . 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 283	. . . 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 257	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
19	5	sscls 22943	. . . . 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 1138	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌)
22	20, 21	ssind 4204	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
23		eqid 2729	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
24	23	clsss2 22959	. . 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 6861	. . . . . 6 ⊢ (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌))
27	26	fveq1i 6859	. . . . 5 ⊢ ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆)
28		id 22	. . . . . . . . 9 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
29	5	topopn 22793	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
30		ssexg 5278	. . . . . . . . 9 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
31	28, 29, 30	syl2anr 597	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
32		resttop 23047	. . . . . . . 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 1132	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top)
35	5	restuni 23049	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
36	35	3adant3 1132	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ (𝐽 ↾t 𝑌))
37	21, 36	sseqtrd 3983	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌))
38		eqid 2729	. . . . . . 7 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ (𝐽 ↾t 𝑌)
39	38	clscld 22934	. . . . . 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	eqeltrid 2832	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)))
42	5	restcld 23059	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌)))
43	42	3adant3 1132	. . . 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 232	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))
45	12, 33	eqeltrid 2832	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
46	45	3adant3 1132	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top)
47	12	unieqi 4883	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)
48	47	eqcomi 2738	. . . . . . . 8 ⊢ ∪ (𝐽 ↾t 𝑌) = ∪ 𝐾
49	48	sscls 22943	. . . . . . 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 480	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
52		inss1 4200	. . . . . . 7 ⊢ (𝑥 ∩ 𝑌) ⊆ 𝑥
53		sseq1 3972	. . . . . . 7 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ 𝑌) ⊆ 𝑥))
54	52, 53	mpbiri 258	. . . . . 6 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
55	54	ad2antll 729	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
56	51, 55	sstrd 3957	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ 𝑥)
57	5	clsss2 22959	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
58	57	adantl 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
59	58	ssrind 4207	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌))
60		sseq2 3973	. . . . . . . 8 ⊢ (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌)))
61	59, 60	syl5ibrcom 247	. . . . . . 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 456	. . . . . 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 454	. . . 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 3140	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
67	25, 66	eqssd 3964	1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  Clsdccld 22903  clsccl 22905

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-iin 4958  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  df-cls 22908

This theorem is referenced by:  restlp  23070  resscdrg  25258  restcls2  48902