Theorem cldbnd 36947

Description: A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)

Hypothesis

Ref	Expression
opnbnd.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldbnd	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))

Proof of Theorem cldbnd

Step	Hyp	Ref	Expression
1		opnbnd.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	iscld3 23293	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3		eqimss 3992	. . . 4 ⊢ (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
4	2, 3	biimtrdi 256	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5		ssinss1 4194	. . 3 ⊢ (((cls‘𝐽)‘𝐴) ⊆ 𝐴 → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴)
6	4, 5	syl6 36	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))
7		sslin 4191	. . . . . 6 ⊢ ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴))
8	7	adantl 487	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴))
9		disjdifr 4430	. . . . 5 ⊢ ((𝑋 ∖ 𝐴) ∩ 𝐴) = ∅
10		sseq0 4357	. . . . 5 ⊢ ((((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴) ∧ ((𝑋 ∖ 𝐴) ∩ 𝐴) = ∅) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅)
11	8, 9, 10	sylancl 598	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅)
12	11	ex 418	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅))
13		incom 4158	. . . . . . . 8 ⊢ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘𝐴))
14		dfss4 4218	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		fveq2 6882	. . . . . . . . . . . 12 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))) = ((cls‘𝐽)‘𝐴))
16	15	eqcomd 2768	. . . . . . . . . . 11 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
17	14, 16	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
18	17	adantl 487	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
19	18	ineq2d 4169	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
20	13, 19	eqtrid 2809	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
21	20	ineq2d 4169	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))))
22	21	eqeq1d 2764	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
23		difss 4086	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
24	1	opnbnd 36946	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
25	23, 24	mpan2 704	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
26	25	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
27	22, 26	bitr4d 285	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
28	1	opncld 23262	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽))
29	28	ex 418	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽)))
30	29	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽)))
31		eleq1 2850	. . . . . . 7 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
32	14, 31	sylbi 220	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
33	32	adantl 487	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
34	30, 33	sylibd 242	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 → 𝐴 ∈ (Clsd‘𝐽)))
35	27, 34	sylbid 243	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ → 𝐴 ∈ (Clsd‘𝐽)))
36	12, 35	syld 48	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → 𝐴 ∈ (Clsd‘𝐽)))
37	6, 36	impbid 215	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ‘cfv 6537  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-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-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-id 5554  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-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23123  df-cld 23248  df-ntr 23249  df-cls 23250

This theorem is used by: (None)