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Theorem ntrss 22946
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 4134 . . . . . . 7 (𝑇 βŠ† 𝑆 β†’ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))
21adantl 481 . . . . . 6 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ (𝑋 βˆ– 𝑆) βŠ† (𝑋 βˆ– 𝑇))
3 difss 4127 . . . . . 6 (𝑋 βˆ– 𝑇) βŠ† 𝑋
42, 3jctil 519 . . . . 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 22945 . . . . . 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 1118 . . . . 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 592 . . . 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 4137 . . 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 3985 . . . . 5 (𝑇 βŠ† 𝑆 β†’ (𝑆 βŠ† 𝑋 β†’ 𝑇 βŠ† 𝑋))
1110impcom 407 . . . 4 ((𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆) β†’ 𝑇 βŠ† 𝑋)
125ntrval2 22942 . . . 4 ((𝐽 ∈ Top ∧ 𝑇 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘‡) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑇))))
1311, 12sylan2 592 . . 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 22942 . . . 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 4025 . 2 ((𝐽 ∈ Top ∧ (𝑆 βŠ† 𝑋 ∧ 𝑇 βŠ† 𝑆)) β†’ ((intβ€˜π½)β€˜π‘‡) βŠ† ((intβ€˜π½)β€˜π‘†))
17163impb 1113 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   βˆ– cdif 3941   βŠ† wss 3944  βˆͺ cuni 4903  β€˜cfv 6542  Topctop 22782  intcnt 22908  clsccl 22909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22783  df-cld 22910  df-ntr 22911  df-cls 22912
This theorem is referenced by:  ntrin  22952  ntrcls0  22967  dvreslem  25825  dvres2lem  25826  dvaddbr  25855  dvmulbr  25856  dvmulbrOLD  25857  dvcnvrelem2  25938  ntruni  35747  cldregopn  35751  limciccioolb  44932  limcicciooub  44948  cncfiooicclem1  45204
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