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Theorem ntrss 19422
Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrss  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  T )  C_  ( ( int `  J
) `  S )
)

Proof of Theorem ntrss
StepHypRef Expression
1 sscon 3620 . . . . . . 7  |-  ( T 
C_  S  ->  ( X  \  S )  C_  ( X  \  T ) )
21adantl 466 . . . . . 6  |-  ( ( S  C_  X  /\  T  C_  S )  -> 
( X  \  S
)  C_  ( X  \  T ) )
3 difss 3613 . . . . . 6  |-  ( X 
\  T )  C_  X
42, 3jctil 537 . . . . 5  |-  ( ( S  C_  X  /\  T  C_  S )  -> 
( ( X  \  T )  C_  X  /\  ( X  \  S
)  C_  ( X  \  T ) ) )
5 clscld.1 . . . . . . 7  |-  X  = 
U. J
65clsss 19421 . . . . . 6  |-  ( ( J  e.  Top  /\  ( X  \  T ) 
C_  X  /\  ( X  \  S )  C_  ( X  \  T ) )  ->  ( ( cls `  J ) `  ( X  \  S ) )  C_  ( ( cls `  J ) `  ( X  \  T ) ) )
763expb 1196 . . . . 5  |-  ( ( J  e.  Top  /\  ( ( X  \  T )  C_  X  /\  ( X  \  S
)  C_  ( X  \  T ) ) )  ->  ( ( cls `  J ) `  ( X  \  S ) ) 
C_  ( ( cls `  J ) `  ( X  \  T ) ) )
84, 7sylan2 474 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( cls `  J ) `  ( X  \  S ) ) 
C_  ( ( cls `  J ) `  ( X  \  T ) ) )
98sscond 3623 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) 
C_  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
10 sstr2 3493 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
1110impcom 430 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
125ntrval2 19418 . . . 4  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( int `  J
) `  T )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) )
1311, 12sylan2 474 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  =  ( X 
\  ( ( cls `  J ) `  ( X  \  T ) ) ) )
145ntrval2 19418 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
1514adantrr 716 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  S
)  =  ( X 
\  ( ( cls `  J ) `  ( X  \  S ) ) ) )
169, 13, 153sstr4d 3529 . 2  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  C_  ( ( int `  J ) `  S ) )
17163impb 1191 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  T )  C_  ( ( int `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    \ cdif 3455    C_ wss 3458   U.cuni 4230   ` cfv 5574   Topctop 19261   intcnt 19384   clsccl 19385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-top 19266  df-cld 19386  df-ntr 19387  df-cls 19388
This theorem is referenced by:  ntrin  19428  ntrcls0  19443  dvreslem  22179  dvres2lem  22180  dvaddbr  22207  dvmulbr  22208  dvcnvrelem2  22285  ntruni  30113  cldregopn  30117  limciccioolb  31531  limcicciooub  31547  cncfiooicclem1  31599
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