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Theorem ntrval2 21057
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 3880 . . . . . 6 (𝑋𝑆) ⊆ 𝑋
2 clscld.1 . . . . . . 7 𝑋 = 𝐽
32clsval2 21056 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
41, 3mpan2 709 . . . . 5 (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
54adantr 472 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
6 dfss4 4001 . . . . . . . 8 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76biimpi 206 . . . . . . 7 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
87fveq2d 6356 . . . . . 6 (𝑆𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
98adantl 473 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
109difeq2d 3871 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
115, 10eqtrd 2794 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
1211difeq2d 3871 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
132ntropn 21055 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
142eltopss 20914 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
1513, 14syldan 488 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16 dfss4 4001 . . 3 (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1715, 16sylib 208 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1812, 17eqtr2d 2795 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cdif 3712  wss 3715   cuni 4588  cfv 6049  Topctop 20900  intcnt 21023  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-ntr 21026  df-cls 21027
This theorem is referenced by:  ntrdif  21058  ntrss  21061  kur14lem2  31496  dssmapntrcls  38928
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