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Theorem opnbnd 31296
Description: A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
Hypothesis
Ref Expression
opnbnd.1 𝑋 = 𝐽
Assertion
Ref Expression
opnbnd ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))

Proof of Theorem opnbnd
StepHypRef Expression
1 disjdif 3991 . . . . 5 (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅
21a1i 11 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
3 ineq1 3768 . . . . 5 (((int‘𝐽)‘𝐴) = 𝐴 → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
43eqeq1d 2611 . . . 4 (((int‘𝐽)‘𝐴) = 𝐴 → ((((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
52, 4syl5ibcom 233 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
6 opnbnd.1 . . . . . . 7 𝑋 = 𝐽
76ntrss2 20613 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
87adantr 479 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
9 inssdif0 3900 . . . . . 6 ((𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴) ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
106sscls 20612 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
11 df-ss 3553 . . . . . . . . . 10 (𝐴 ⊆ ((cls‘𝐽)‘𝐴) ↔ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
1210, 11sylib 206 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
1312eqcomd 2615 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
14 eqimss 3619 . . . . . . . 8 (𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)) → 𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
1513, 14syl 17 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
16 sstr 3575 . . . . . . 7 ((𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
1715, 16sylan 486 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
189, 17sylan2br 491 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
198, 18eqssd 3584 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) = 𝐴)
2019ex 448 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ → ((int‘𝐽)‘𝐴) = 𝐴))
215, 20impbid 200 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
226isopn3 20622 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ ((int‘𝐽)‘𝐴) = 𝐴))
236topbnd 31295 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
2423ineq2d 3775 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
2524eqeq1d 2611 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
2621, 22, 253bitr4d 298 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  cdif 3536  cin 3538  wss 3539  c0 3873   cuni 4366  cfv 5790  Topctop 20459  intcnt 20573  clsccl 20574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-top 20463  df-cld 20575  df-ntr 20576  df-cls 20577
This theorem is referenced by:  cldbnd  31297
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