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Theorem clsdif 21758
 Description: A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
clsdif ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))

Proof of Theorem clsdif
StepHypRef Expression
1 difss 4039 . . . 4 (𝑋𝐴) ⊆ 𝑋
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval2 21755 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
41, 3mpan2 690 . . 3 (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
54adantr 484 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
6 simpr 488 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
7 dfss4 4165 . . . . 5 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
86, 7sylib 221 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
98fveq2d 6666 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((int‘𝐽)‘𝐴))
109difeq2d 4030 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
115, 10eqtrd 2793 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∖ cdif 3857   ⊆ wss 3860  ∪ cuni 4801  ‘cfv 6339  Topctop 21598  intcnt 21722  clsccl 21723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-top 21599  df-cld 21724  df-ntr 21725  df-cls 21726 This theorem is referenced by:  maxlp  21852  topbnd  34088
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