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Theorem cldbnd 32909
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 21276 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝐴) = 𝐴))
3 eqimss 3876 . . . 4 (((cls‘𝐽)‘𝐴) = 𝐴 → ((cls‘𝐽)‘𝐴) ⊆ 𝐴)
42, 3syl6bi 245 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐴))
5 ssinss1 4062 . . 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 4059 . . . . . 6 ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴))
87adantl 475 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴))
9 incom 4028 . . . . . 6 ((𝑋𝐴) ∩ 𝐴) = (𝐴 ∩ (𝑋𝐴))
10 disjdif 4264 . . . . . 6 (𝐴 ∩ (𝑋𝐴)) = ∅
119, 10eqtri 2802 . . . . 5 ((𝑋𝐴) ∩ 𝐴) = ∅
12 sseq0 4201 . . . . 5 ((((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) ⊆ ((𝑋𝐴) ∩ 𝐴) ∧ ((𝑋𝐴) ∩ 𝐴) = ∅) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
138, 11, 12sylancl 580 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅)
1413ex 403 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴 → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
15 incom 4028 . . . . . . . 8 (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴))
16 dfss4 4085 . . . . . . . . . . 11 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
17 fveq2 6446 . . . . . . . . . . . 12 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((cls‘𝐽)‘𝐴))
1817eqcomd 2784 . . . . . . . . . . 11 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
1916, 18sylbi 209 . . . . . . . . . 10 (𝐴𝑋 → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
2019adantl 475 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) = ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))
2120ineq2d 4037 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘𝐴)) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2215, 21syl5eq 2826 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
2322ineq2d 4037 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))))
2423eqeq1d 2780 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
25 difss 3960 . . . . . . 7 (𝑋𝐴) ⊆ 𝑋
261opnbnd 32908 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2725, 26mpan2 681 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2827adantr 474 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 ↔ ((𝑋𝐴) ∩ (((cls‘𝐽)‘(𝑋𝐴)) ∩ ((cls‘𝐽)‘(𝑋 ∖ (𝑋𝐴))))) = ∅))
2924, 28bitr4d 274 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((𝑋𝐴) ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ (𝑋𝐴) ∈ 𝐽))
301opncld 21245 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝐴) ∈ 𝐽) → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽))
3130ex 403 . . . . . 6 (𝐽 ∈ Top → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
3231adantr 474 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽 → (𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽)))
33 eleq1 2847 . . . . . . 7 ((𝑋 ∖ (𝑋𝐴)) = 𝐴 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3416, 33sylbi 209 . . . . . 6 (𝐴𝑋 → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3534adantl 475 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋 ∖ (𝑋𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
3632, 35sylibd 231 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝑋𝐴) ∈ 𝐽𝐴 ∈ (Clsd‘𝐽)))
3729, 36sylbid 232 . . 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 204 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 198  wa 386   = wceq 1601  wcel 2107  cdif 3789  cin 3791  wss 3792  c0 4141   cuni 4671  cfv 6135  Topctop 21105  Clsdccld 21228  clsccl 21230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-top 21106  df-cld 21231  df-ntr 21232  df-cls 21233
This theorem is referenced by: (None)
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