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Theorem clsdif 20609
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 3698 . . . 4 (𝑋𝐴) ⊆ 𝑋
2 clscld.1 . . . . 5 𝑋 = 𝐽
32clsval2 20606 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝐴) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
41, 3mpan2 702 . . 3 (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
54adantr 479 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))))
6 simpr 475 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴𝑋)
7 dfss4 3819 . . . . 5 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
86, 7sylib 206 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
98fveq2d 6092 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴))) = ((int‘𝐽)‘𝐴))
109difeq2d 3689 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝐴)))) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
115, 10eqtrd 2643 1 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘(𝑋𝐴)) = (𝑋 ∖ ((int‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cdif 3536  wss 3539   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:  maxlp  20703  topbnd  31295
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