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Theorem cldbnd 35712
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 ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))

Proof of Theorem cldbnd
StepHypRef Expression
1 opnbnd.1 . . . . 5 𝑋 = 𝐽
21iscld3 22912 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3 eqimss 4033 . . . 4 (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
42, 3syl6bi 253 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5 ssinss1 4230 . . 3 (((cls‘𝐽)‘𝐴) ⊆ 𝐴 → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴)
64, 5syl6 35 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))
7 sslin 4227 . . . . . 6 ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴))
87adantl 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴))
9 disjdifr 4465 . . . . 5 ((𝑋𝐴) ∩ 𝐴) = ∅
10 sseq0 4392 . . . . 5 ((((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴) ∧ ((𝑋𝐴) ∩ 𝐴) = ∅) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
118, 9, 10sylancl 585 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
1211ex 412 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
13 incom 4194 . . . . . . . 8 (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴))
14 dfss4 4251 . . . . . . . . . . 11 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
15 fveq2 6882 . . . . . . . . . . . 12 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((cls‘𝐽)‘𝐴))
1615eqcomd 2730 . . . . . . . . . . 11 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1714, 16sylbi 216 . . . . . . . . . 10 (𝐴𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1817adantl 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1918ineq2d 4205 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2013, 19eqtrid 2776 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2120ineq2d 4205 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))))
2221eqeq1d 2726 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
23 difss 4124 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
241opnbnd 35711 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2523, 24mpan2 688 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2625adantr 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2722, 26bitr4d 282 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ (𝑋𝐴) ∈ 𝐽))
281opncld 22881 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽))
2928ex 412 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
3029adantr 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
31 eleq1 2813 . . . . . . 7 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3214, 31sylbi 216 . . . . . 6 (𝐴𝑋 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3332adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3430, 33sylibd 238 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽𝐴 ∈ (Clsd‘𝐽)))
3527, 34sylbid 239 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ → 𝐴 ∈ (Clsd‘𝐽)))
3612, 35syld 47 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴𝐴 ∈ (Clsd‘𝐽)))
376, 36impbid 211 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cdif 3938  cin 3940  wss 3941  c0 4315   cuni 4900  cfv 6534  Topctop 22739  Clsdccld 22864  clsccl 22866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-top 22740  df-cld 22867  df-ntr 22868  df-cls 22869
This theorem is referenced by: (None)
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