MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ntrss Structured version   Visualization version   GIF version

Theorem ntrss 22150
Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
ntrss ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))

Proof of Theorem ntrss
StepHypRef Expression
1 sscon 4074 . . . . . . 7 (𝑇𝑆 → (𝑋𝑆) ⊆ (𝑋𝑇))
21adantl 481 . . . . . 6 ((𝑆𝑋𝑇𝑆) → (𝑋𝑆) ⊆ (𝑋𝑇))
3 difss 4067 . . . . . 6 (𝑋𝑇) ⊆ 𝑋
42, 3jctil 519 . . . . 5 ((𝑆𝑋𝑇𝑆) → ((𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇)))
5 clscld.1 . . . . . . 7 𝑋 = 𝐽
65clsss 22149 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇)) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
763expb 1118 . . . . 5 ((𝐽 ∈ Top ∧ ((𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇))) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
84, 7sylan2 592 . . . 4 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
98sscond 4077 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))) ⊆ (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
10 sstr2 3929 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
1110impcom 407 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
125ntrval2 22146 . . . 4 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((int‘𝐽)‘𝑇) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))))
1311, 12sylan2 592 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑇) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))))
145ntrval2 22146 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
1514adantrr 713 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
169, 13, 153sstr4d 3969 . 2 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
17163impb 1113 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1539  wcel 2107  cdif 3885  wss 3888   cuni 4841  cfv 6423  Topctop 21986  intcnt 22112  clsccl 22113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  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 2708  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7571
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3429  df-sbc 3717  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-int 4882  df-iun 4928  df-iin 4929  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6381  df-fun 6425  df-fn 6426  df-f 6427  df-f1 6428  df-fo 6429  df-f1o 6430  df-fv 6431  df-top 21987  df-cld 22114  df-ntr 22115  df-cls 22116
This theorem is referenced by:  ntrin  22156  ntrcls0  22171  dvreslem  25016  dvres2lem  25017  dvaddbr  25045  dvmulbr  25046  dvcnvrelem2  25125  ntruni  34485  cldregopn  34489  limciccioolb  43094  limcicciooub  43110  cncfiooicclem1  43366
  Copyright terms: Public domain W3C validator