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Theorem cldbnd 36309
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 23088 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3 eqimss 4054 . . . 4 (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
42, 3biimtrdi 253 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5 ssinss1 4254 . . 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 4251 . . . . . 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 4479 . . . . 5 ((𝑋𝐴) ∩ 𝐴) = ∅
10 sseq0 4409 . . . . 5 ((((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴) ∧ ((𝑋𝐴) ∩ 𝐴) = ∅) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
118, 9, 10sylancl 586 . . . 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 4217 . . . . . . . 8 (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴))
14 dfss4 4275 . . . . . . . . . . 11 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
15 fveq2 6907 . . . . . . . . . . . 12 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((cls‘𝐽)‘𝐴))
1615eqcomd 2741 . . . . . . . . . . 11 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1714, 16sylbi 217 . . . . . . . . . 10 (𝐴𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1817adantl 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1918ineq2d 4228 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2013, 19eqtrid 2787 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2120ineq2d 4228 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))))
2221eqeq1d 2737 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
23 difss 4146 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
241opnbnd 36308 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2523, 24mpan2 691 . . . . . 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 23057 . . . . . . 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 2827 . . . . . . 7 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3214, 31sylbi 217 . . . . . 6 (𝐴𝑋 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3332adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3430, 33sylibd 239 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽𝐴 ∈ (Clsd‘𝐽)))
3527, 34sylbid 240 . . 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 212 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 206  wa 395   = wceq 1537  wcel 2106  cdif 3960  cin 3962  wss 3963  c0 4339   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  clsccl 23042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-top 22916  df-cld 23043  df-ntr 23044  df-cls 23045
This theorem is referenced by: (None)
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