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Theorem ntrval2 21274
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 3960 . . . . . 6 (𝑋𝑆) ⊆ 𝑋
2 clscld.1 . . . . . . 7 𝑋 = 𝐽
32clsval2 21273 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
41, 3mpan2 681 . . . . 5 (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
54adantr 474 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
6 dfss4 4085 . . . . . . . 8 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76biimpi 208 . . . . . . 7 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
87fveq2d 6452 . . . . . 6 (𝑆𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
98adantl 475 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
109difeq2d 3951 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
115, 10eqtrd 2814 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
1211difeq2d 3951 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
132ntropn 21272 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
142eltopss 21130 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
1513, 14syldan 585 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16 dfss4 4085 . . 3 (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1715, 16sylib 210 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1812, 17eqtr2d 2815 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  cdif 3789  wss 3792   cuni 4673  cfv 6137  Topctop 21116  intcnt 21240  clsccl 21241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-top 21117  df-cld 21242  df-ntr 21243  df-cls 21244
This theorem is referenced by:  ntrdif  21275  ntrss  21278  kur14lem2  31796  dssmapntrcls  39396
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