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Theorem ntrss 22989
Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ntrss ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((intβ€˜π½)β€˜π‘‡) βŠ† ((intβ€˜π½)β€˜π‘†))
Proof of Theorem ntrss
StepHypRef Expression
1 sscon 4136 . . . . . . 7 (𝑇 βŠ† 𝑆 β†’ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))
21adantl 480 . . . . . 6 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))
3 difss 4129 . . . . . 6 (𝑋 βˆ– 𝑇) βŠ† 𝑋
42, 3jctil 518 . . . . 5 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((𝑋 βˆ– 𝑇) βŠ† 𝑋 ∧ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇)))
5 clscld.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
65clsss 22988 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋 βˆ– 𝑇) βŠ† 𝑋 ∧ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇)) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) βŠ† ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇)))
763expb 1117 . . . . 5 ((𝐽 ∈ Top ∧ ((𝑋 βˆ– 𝑇) βŠ† 𝑋 ∧ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) βŠ† ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇)))
84, 7sylan2 591 . . . 4 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆)) βŠ† ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇)))
98sscond 4139 . . 3 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇))) βŠ† (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
10 sstr2 3984 . . . . 5 (𝑇 βŠ† 𝑆 β†’ (𝑆 βŠ† 𝑋 β†’ 𝑇 βŠ† 𝑋))
1110impcom 406 . . . 4 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑋)
125ntrval2 22985 . . . 4 ((𝐽 ∈ Top ∧ 𝑇 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘‡) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇))))
1311, 12sylan2 591 . . 3 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ ((intβ€˜π½)β€˜π‘‡) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇))))
145ntrval2 22985 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
1514adantrr 715 . . 3 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ ((intβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
169, 13, 153sstr4d 4025 . 2 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ ((intβ€˜π½)β€˜π‘‡) βŠ† ((intβ€˜π½)β€˜π‘†))
17163impb 1112 1 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ ((intβ€˜π½)β€˜π‘‡) βŠ† ((intβ€˜π½)β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3942   βŠ† wss 3945  βˆͺ cuni 4908  β€˜cfv 6547  Topctop 22825  intcnt 22951  clsccl 22952
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
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 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-top 22826  df-cld 22953  df-ntr 22954  df-cls 22955
This theorem is referenced by:  ntrin  22995  ntrcls0  23010  dvreslem  25868  dvres2lem  25869  dvaddbr  25898  dvmulbr  25899  dvmulbrOLD  25900  dvcnvrelem2  25981  ntruni  35881  cldregopn  35885  limciccioolb  45072  limcicciooub  45088  cncfiooicclem1  45344
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