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Theorem opnbnd 31490
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 3992 . . . . 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 3769 . . . . 5 (((int‘𝐽)‘𝐴) = 𝐴 → (((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
43eqeq1d 2612 . . . 4 (((int‘𝐽)‘𝐴) = 𝐴 → ((((int‘𝐽)‘𝐴) ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
52, 4syl5ibcom 234 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
6 opnbnd.1 . . . . . . 7 𝑋 = 𝐽
76ntrss2 20671 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
87adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
9 inssdif0 3901 . . . . . 6 ((𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴) ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅)
106sscls 20670 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
11 df-ss 3554 . . . . . . . . . 10 (𝐴 ⊆ ((cls‘𝐽)‘𝐴) ↔ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
1210, 11sylib 207 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∩ ((cls‘𝐽)‘𝐴)) = 𝐴)
1312eqcomd 2616 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐴 ∩ ((cls‘𝐽)‘𝐴)))
14 eqimss 3620 . . . . . . . 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 3576 . . . . . . 7 ((𝐴 ⊆ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
1715, 16sylan 487 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((int‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
189, 17sylan2br 492 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → 𝐴 ⊆ ((int‘𝐽)‘𝐴))
198, 18eqssd 3585 . . . 4 (((𝐽 ∈ Top ∧ 𝐴𝑋) ∧ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅) → ((int‘𝐽)‘𝐴) = 𝐴)
2019ex 449 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅ → ((int‘𝐽)‘𝐴) = 𝐴))
215, 20impbid 201 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((int‘𝐽)‘𝐴) = 𝐴 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
226isopn3 20680 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ ((int‘𝐽)‘𝐴) = 𝐴))
236topbnd 31489 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))
2423ineq2d 3776 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))))
2524eqeq1d 2612 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅ ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴))) = ∅))
2621, 22, 253bitr4d 299 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cdif 3537  cin 3539  wss 3540  c0 3874   cuni 4372  cfv 5804  Topctop 20517  intcnt 20631  clsccl 20632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635
This theorem is referenced by:  cldbnd  31491
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