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Theorem cmclsopn 20676
 Description: The complement of a closure is open. (Contributed by NM, 11-Sep-2006.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cmclsopn ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)

Proof of Theorem cmclsopn
StepHypRef Expression
1 clscld.1 . . . 4 𝑋 = 𝐽
21clsval2 20664 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆))))
32difeq2d 3690 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))))
4 difss 3699 . . . . . . 7 (𝑋𝑆) ⊆ 𝑋
51ntropn 20663 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
64, 5mpan2 703 . . . . . 6 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽)
71eltopss 20537 . . . . . 6 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋𝑆)) ∈ 𝐽) → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
86, 7mpdan 699 . . . . 5 (𝐽 ∈ Top → ((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋)
9 dfss4 3820 . . . . 5 (((int‘𝐽)‘(𝑋𝑆)) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
108, 9sylib 207 . . . 4 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) = ((int‘𝐽)‘(𝑋𝑆)))
1110, 6eqeltrd 2688 . . 3 (𝐽 ∈ Top → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
1211adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘(𝑋𝑆)))) ∈ 𝐽)
133, 12eqeltrd 2688 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘𝑆)) ∈ 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∪ 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:  elcls  20687
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