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Theorem cldbnd 32446
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 20916 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3 eqimss 3690 . . . 4 (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
42, 3syl6bi 243 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5 ssinss1 3874 . . 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 3872 . . . . . 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 incom 3838 . . . . . 6 ((𝑋𝐴) ∩ 𝐴) = (𝐴 ∩ (𝑋𝐴))
10 disjdif 4073 . . . . . 6 (𝐴 ∩ (𝑋𝐴)) = ∅
119, 10eqtri 2673 . . . . 5 ((𝑋𝐴) ∩ 𝐴) = ∅
12 sseq0 4008 . . . . 5 ((((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴) ∧ ((𝑋𝐴) ∩ 𝐴) = ∅) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
138, 11, 12sylancl 695 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
1413ex 449 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
15 incom 3838 . . . . . . . 8 (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴))
16 dfss4 3891 . . . . . . . . . . 11 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
17 fveq2 6229 . . . . . . . . . . . 12 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((cls‘𝐽)‘𝐴))
1817eqcomd 2657 . . . . . . . . . . 11 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1916, 18sylbi 207 . . . . . . . . . 10 (𝐴𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
2019adantl 481 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
2120ineq2d 3847 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2215, 21syl5eq 2697 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2322ineq2d 3847 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))))
2423eqeq1d 2653 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
25 difss 3770 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
261opnbnd 32445 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2725, 26mpan2 707 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2827adantr 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2924, 28bitr4d 271 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ (𝑋𝐴) ∈ 𝐽))
301opncld 20885 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽))
3130ex 449 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
3231adantr 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
33 eleq1 2718 . . . . . . 7 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3416, 33sylbi 207 . . . . . 6 (𝐴𝑋 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3534adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3632, 35sylibd 229 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽𝐴 ∈ (Clsd‘𝐽)))
3729, 36sylbid 230 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ → 𝐴 ∈ (Clsd‘𝐽)))
3814, 37syld 47 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴𝐴 ∈ (Clsd‘𝐽)))
396, 38impbid 202 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 196  wa 383   = wceq 1523  wcel 2030  cdif 3604  cin 3606  wss 3607  c0 3948   cuni 4468  cfv 5926  Topctop 20746  Clsdccld 20868  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by: (None)
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