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Theorem cldbnd 31960
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 20778 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3 eqimss 3636 . . . 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 3819 . . 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 3817 . . . . . 6 ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴))
87adantl 482 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴))
9 incom 3783 . . . . . 6 ((𝑋𝐴) ∩ 𝐴) = (𝐴 ∩ (𝑋𝐴))
10 disjdif 4012 . . . . . 6 (𝐴 ∩ (𝑋𝐴)) = ∅
119, 10eqtri 2643 . . . . 5 ((𝑋𝐴) ∩ 𝐴) = ∅
12 sseq0 3947 . . . . 5 ((((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴) ∧ ((𝑋𝐴) ∩ 𝐴) = ∅) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
138, 11, 12sylancl 693 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
1413ex 450 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
15 incom 3783 . . . . . . . 8 (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴))
16 dfss4 3836 . . . . . . . . . . 11 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
17 fveq2 6148 . . . . . . . . . . . 12 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((cls‘𝐽)‘𝐴))
1817eqcomd 2627 . . . . . . . . . . 11 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1916, 18sylbi 207 . . . . . . . . . 10 (𝐴𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
2019adantl 482 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
2120ineq2d 3792 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2215, 21syl5eq 2667 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2322ineq2d 3792 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))))
2423eqeq1d 2623 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
25 difss 3715 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
261opnbnd 31959 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2725, 26mpan2 706 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2827adantr 481 . . . . 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 20747 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽))
3130ex 450 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
3231adantr 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
33 eleq1 2686 . . . . . . 7 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3416, 33sylbi 207 . . . . . 6 (𝐴𝑋 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3534adantl 482 . . . . 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 384   = wceq 1480  wcel 1987  cdif 3552  cin 3554  wss 3555  c0 3891   cuni 4402  cfv 5847  Topctop 20617  Clsdccld 20730  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-cld 20733  df-ntr 20734  df-cls 20735
This theorem is referenced by: (None)
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