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Theorem ntrss 18792
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 3599 . . . . . . 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 3592 . . . . . 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 18791 . . . . . 6  |-  ( ( J  e.  Top  /\  ( X  \  T ) 
C_  X  /\  ( X  \  S )  C_  ( X  \  T ) )  ->  ( ( cls `  J ) `  ( X  \  S ) )  C_  ( ( cls `  J ) `  ( X  \  T ) ) )
763expb 1189 . . . . 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 3602 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) 
C_  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
10 sstr2 3472 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
1110impcom 430 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
125ntrval2 18788 . . . 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 18788 . . . 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 3508 . 2  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  C_  ( ( int `  J ) `  S ) )
17163impb 1184 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 965    = wceq 1370    e. wcel 1758    \ cdif 3434    C_ wss 3437   U.cuni 4200   ` cfv 5527   Topctop 18631   intcnt 18754   clsccl 18755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-top 18636  df-cld 18756  df-ntr 18757  df-cls 18758
This theorem is referenced by:  ntrin  18798  ntrcls0  18813  dvreslem  21518  dvres2lem  21519  dvaddbr  21546  dvmulbr  21547  dvcnvrelem2  21624  ntruni  28671  cldregopn  28675
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