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Theorem ntrval2 22876
Description: Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ntrval2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
Proof of Theorem ntrval2
StepHypRef Expression
1 difss 4131 . . . . . 6 (𝑋 βˆ– 𝑆) βŠ† 𝑋
2 clscld.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
32clsval2 22875 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋 βˆ– 𝑆) βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– (𝑋 βˆ– 𝑆)))))
41, 3mpan2 688 . . . . 5 (𝐽 ∈ Top β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– (𝑋 βˆ– 𝑆)))))
54adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– (𝑋 βˆ– 𝑆)))))
6 dfss4 4258 . . . . . . . 8 (𝑆 βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– 𝑆)) = 𝑆)
76biimpi 215 . . . . . . 7 (𝑆 βŠ† 𝑋 β†’ (𝑋 βˆ– (𝑋 βˆ– 𝑆)) = 𝑆)
87fveq2d 6895 . . . . . 6 (𝑆 βŠ† 𝑋 β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– (𝑋 βˆ– 𝑆))) = ((intβ€˜π½)β€˜π‘†))
98adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(𝑋 βˆ– (𝑋 βˆ– 𝑆))) = ((intβ€˜π½)β€˜π‘†))
109difeq2d 4122 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑋 βˆ– ((intβ€˜π½)β€˜(𝑋 βˆ– (𝑋 βˆ– 𝑆)))) = (𝑋 βˆ– ((intβ€˜π½)β€˜π‘†)))
115, 10eqtrd 2771 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) = (𝑋 βˆ– ((intβ€˜π½)β€˜π‘†)))
1211difeq2d 4122 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))) = (𝑋 βˆ– (𝑋 βˆ– ((intβ€˜π½)β€˜π‘†))))
132ntropn 22874 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽)
142eltopss 22730 . . . 4 ((𝐽 ∈ Top ∧ ((intβ€˜π½)β€˜π‘†) ∈ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑋)
1513, 14syldan 590 . . 3 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑋)
16 dfss4 4258 . . 3 (((intβ€˜π½)β€˜π‘†) βŠ† 𝑋 ↔ (𝑋 βˆ– (𝑋 βˆ– ((intβ€˜π½)β€˜π‘†))) = ((intβ€˜π½)β€˜π‘†))
1715, 16sylib 217 . 2 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ (𝑋 βˆ– (𝑋 βˆ– ((intβ€˜π½)β€˜π‘†))) = ((intβ€˜π½)β€˜π‘†))
1812, 17eqtr2d 2772 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βˆ– cdif 3945   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22716  intcnt 22842  clsccl 22843
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22717  df-cld 22844  df-ntr 22845  df-cls 22846
This theorem is referenced by:  ntrdif  22877  ntrss  22880  kur14lem2  34664  dssmapntrcls  43345
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