Theorem cldbnd 36381

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 22989	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3		eqimss 3990	. . . 4 ⊢ (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
4	2, 3	biimtrdi 253	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5		ssinss1 4197	. . 3 ⊢ (((cls‘𝐽)‘𝐴) ⊆ 𝐴 → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴)
6	4, 5	syl6 35	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))
7		sslin 4194	. . . . . 6 ⊢ ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) ⊆ ((𝑋 ∖ 𝐴) ∩ 𝐴))
8	7	adantl 481	. . . . 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 4424	. . . . 5 ⊢ ((𝑋 ∖ 𝐴) ∩ 𝐴) = ∅
10		sseq0 4354	. . . . 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 586	. . . 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 412	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅))
13		incom 4160	. . . . . . . 8 ⊢ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘𝐴))
14		dfss4 4220	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴)
15		fveq2 6831	. . . . . . . . . . . 12 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))) = ((cls‘𝐽)‘𝐴))
16	15	eqcomd 2739	. . . . . . . . . . 11 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
17	14, 16	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))
19	18	ineq2d 4171	. . . . . . . 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 2780	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) = (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴)))))
21	20	ineq2d 4171	. . . . . 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 2735	. . . . 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 4087	. . . . . . 7 ⊢ (𝑋 ∖ 𝐴) ⊆ 𝑋
24	1	opnbnd 36380	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
25	23, 24	mpan2 691	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
26	25	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 ↔ ((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘(𝑋 ∖ 𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝐴))))) = ∅))
27	22, 26	bitr4d 282	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ ↔ (𝑋 ∖ 𝐴) ∈ 𝐽))
28	1	opncld 22958	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽))
29	28	ex 412	. . . . . 6 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽)))
30	29	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽)))
31		eleq1 2821	. . . . . . 7 ⊢ ((𝑋 ∖ (𝑋 ∖ 𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
32	14, 31	sylbi 217	. . . . . 6 ⊢ (𝐴 ⊆ 𝑋 → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
33	32	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ (𝑋 ∖ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
34	30, 33	sylibd 239	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((𝑋 ∖ 𝐴) ∈ 𝐽 → 𝐴 ∈ (Clsd‘𝐽)))
35	27, 34	sylbid 240	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((𝑋 ∖ 𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) = ∅ → 𝐴 ∈ (Clsd‘𝐽)))
36	12, 35	syld 47	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴 → 𝐴 ∈ (Clsd‘𝐽)))
37	6, 36	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) ⊆ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4284  ∪ cuni 4860  ‘cfv 6489  Topctop 22818  Clsdccld 22941  clsccl 22943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-top 22819  df-cld 22944  df-ntr 22945  df-cls 22946

This theorem is referenced by: (None)