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Theorem ntrss 21591
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 4112 . . . . . . 7 (𝑇𝑆 → (𝑋𝑆) ⊆ (𝑋𝑇))
21adantl 482 . . . . . 6 ((𝑆𝑋𝑇𝑆) → (𝑋𝑆) ⊆ (𝑋𝑇))
3 difss 4105 . . . . . 6 (𝑋𝑇) ⊆ 𝑋
42, 3jctil 520 . . . . 5 ((𝑆𝑋𝑇𝑆) → ((𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇)))
5 clscld.1 . . . . . . 7 𝑋 = 𝐽
65clsss 21590 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑋𝑇) ⊆ 𝑋 ∧ (𝑋𝑆) ⊆ (𝑋𝑇)) → ((cls‘𝐽)‘(𝑋𝑆)) ⊆ ((cls‘𝐽)‘(𝑋𝑇)))
763expb 1112 . . . . 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 4115 . . 3 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑇))) ⊆ (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑆))))
10 sstr2 3971 . . . . 5 (𝑇𝑆 → (𝑆𝑋𝑇𝑋))
1110impcom 408 . . . 4 ((𝑆𝑋𝑇𝑆) → 𝑇𝑋)
125ntrval2 21587 . . . 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 21587 . . . 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 4011 . 2 ((𝐽 ∈ Top ∧ (𝑆𝑋𝑇𝑆)) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
17163impb 1107 1 ((𝐽 ∈ Top ∧ 𝑆𝑋𝑇𝑆) → ((int‘𝐽)‘𝑇) ⊆ ((int‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cdif 3930  wss 3933   cuni 4830  cfv 6348  Topctop 21429  intcnt 21553  clsccl 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-cld 21555  df-ntr 21556  df-cls 21557
This theorem is referenced by:  ntrin  21597  ntrcls0  21612  dvreslem  24434  dvres2lem  24435  dvaddbr  24462  dvmulbr  24463  dvcnvrelem2  24542  ntruni  33572  cldregopn  33576  limciccioolb  41778  limcicciooub  41794  cncfiooicclem1  42052
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