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Theorem ntrval2 22999
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 4128 . . . . . 6 (𝑋𝑆) ⊆ 𝑋
2 clscld.1 . . . . . . 7 𝑋 = 𝐽
32clsval2 22998 . . . . . 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 479 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))))
6 dfss4 4257 . . . . . . . 8 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76biimpi 215 . . . . . . 7 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
87fveq2d 6900 . . . . . 6 (𝑆𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
98adantl 480 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆))) = ((int‘𝐽)‘𝑆))
109difeq2d 4118 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
115, 10eqtrd 2765 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘(𝑋𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆)))
1211difeq2d 4118 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))))
132ntropn 22997 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
142eltopss 22853 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
1513, 14syldan 589 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋)
16 dfss4 4257 . . 3 (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1715, 16sylib 217 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆))
1812, 17eqtr2d 2766 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cdif 3941  wss 3944   cuni 4909  cfv 6549  Topctop 22839  intcnt 22965  clsccl 22966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-top 22840  df-cld 22967  df-ntr 22968  df-cls 22969
This theorem is referenced by:  ntrdif  23000  ntrss  23003  kur14lem2  34948  dssmapntrcls  43700
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