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Theorem ntrval2 21661
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 4110 . . . . . 6 (𝑋𝑆) ⊆ 𝑋
2 clscld.1 . . . . . . 7 𝑋 = 𝐽
32clsval2 21660 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
41, 3mpan2 689 . . . . 5 (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
54adantr 483 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
6 dfss4 4237 . . . . . . . 8 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76biimpi 218 . . . . . . 7 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
87fveq2d 6676 . . . . . 6 (𝑆𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
98adantl 484 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
109difeq2d 4101 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
115, 10eqtrd 2858 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
1211difeq2d 4101 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
132ntropn 21659 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
142eltopss 21517 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
1513, 14syldan 593 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16 dfss4 4237 . . 3 (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1715, 16sylib 220 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1812, 17eqtr2d 2859 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cdif 3935  wss 3938   cuni 4840  cfv 6357  Topctop 21503  intcnt 21627  clsccl 21628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-top 21504  df-cld 21629  df-ntr 21630  df-cls 21631
This theorem is referenced by:  ntrdif  21662  ntrss  21665  kur14lem2  32456  dssmapntrcls  40485
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