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Theorem ntrval2 21795
Description: Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
ntrval2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))

Proof of Theorem ntrval2
StepHypRef Expression
1 difss 4020 . . . . . 6 (𝑋𝑆) ⊆ 𝑋
2 clscld.1 . . . . . . 7 𝑋 = 𝐽
32clsval2 21794 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
41, 3mpan2 691 . . . . 5 (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
54adantr 484 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
6 dfss4 4147 . . . . . . . 8 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76biimpi 219 . . . . . . 7 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
87fveq2d 6672 . . . . . 6 (𝑆𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
98adantl 485 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
109difeq2d 4011 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
115, 10eqtrd 2773 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
1211difeq2d 4011 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
132ntropn 21793 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
142eltopss 21651 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
1513, 14syldan 594 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16 dfss4 4147 . . 3 (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1715, 16sylib 221 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1812, 17eqtr2d 2774 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  cdif 3838  wss 3841   cuni 4793  cfv 6333  Topctop 21637  intcnt 21761  clsccl 21762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-iin 4881  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-top 21638  df-cld 21763  df-ntr 21764  df-cls 21765
This theorem is referenced by:  ntrdif  21796  ntrss  21799  kur14lem2  32732  dssmapntrcls  41268
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