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Theorem ntrss 22559
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 4139 . . . . . . 7 (𝑇 βŠ† 𝑆 β†’ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))
21adantl 483 . . . . . 6 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))
3 difss 4132 . . . . . 6 (𝑋 βˆ– 𝑇) βŠ† 𝑋
42, 3jctil 521 . . . . 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 22558 . . . . . 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 1121 . . . . 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 594 . . . 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 4142 . . 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 3990 . . . . 5 (𝑇 βŠ† 𝑆 β†’ (𝑆 βŠ† 𝑋 β†’ 𝑇 βŠ† 𝑋))
1110impcom 409 . . . 4 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑋)
125ntrval2 22555 . . . 4 ((𝐽 ∈ Top ∧ 𝑇 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘‡) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇))))
1311, 12sylan2 594 . . 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 22555 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘†) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑆))))
1514adantrr 716 . . 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 4030 . 2 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ ((intβ€˜π½)β€˜π‘‡) βŠ† ((intβ€˜π½)β€˜π‘†))
17163impb 1116 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3946   βŠ† wss 3949  βˆͺ cuni 4909  β€˜cfv 6544  Topctop 22395  intcnt 22521  clsccl 22522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  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 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525
This theorem is referenced by:  ntrin  22565  ntrcls0  22580  dvreslem  25426  dvres2lem  25427  dvaddbr  25455  dvmulbr  25456  dvcnvrelem2  25535  gg-dvmulbr  35175  ntruni  35212  cldregopn  35216  limciccioolb  44337  limcicciooub  44353  cncfiooicclem1  44609
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