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Theorem ntrss 19726
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 3624 . . . . . . 7  |-  ( T 
C_  S  ->  ( X  \  S )  C_  ( X  \  T ) )
21adantl 464 . . . . . 6  |-  ( ( S  C_  X  /\  T  C_  S )  -> 
( X  \  S
)  C_  ( X  \  T ) )
3 difss 3617 . . . . . 6  |-  ( X 
\  T )  C_  X
42, 3jctil 535 . . . . 5  |-  ( ( S  C_  X  /\  T  C_  S )  -> 
( ( X  \  T )  C_  X  /\  ( X  \  S
)  C_  ( X  \  T ) ) )
5 clscld.1 . . . . . . 7  |-  X  = 
U. J
65clsss 19725 . . . . . 6  |-  ( ( J  e.  Top  /\  ( X  \  T ) 
C_  X  /\  ( X  \  S )  C_  ( X  \  T ) )  ->  ( ( cls `  J ) `  ( X  \  S ) )  C_  ( ( cls `  J ) `  ( X  \  T ) ) )
763expb 1195 . . . . 5  |-  ( ( J  e.  Top  /\  ( ( X  \  T )  C_  X  /\  ( X  \  S
)  C_  ( X  \  T ) ) )  ->  ( ( cls `  J ) `  ( X  \  S ) ) 
C_  ( ( cls `  J ) `  ( X  \  T ) ) )
84, 7sylan2 472 . . . 4  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( cls `  J ) `  ( X  \  S ) ) 
C_  ( ( cls `  J ) `  ( X  \  T ) ) )
98sscond 3627 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) 
C_  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
10 sstr2 3496 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  X  ->  T  C_  X ) )
1110impcom 428 . . . 4  |-  ( ( S  C_  X  /\  T  C_  S )  ->  T  C_  X )
125ntrval2 19722 . . . 4  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( int `  J
) `  T )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  T ) ) ) )
1311, 12sylan2 472 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  =  ( X 
\  ( ( cls `  J ) `  ( X  \  T ) ) ) )
145ntrval2 19722 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  S ) ) ) )
1514adantrr 714 . . 3  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  S
)  =  ( X 
\  ( ( cls `  J ) `  ( X  \  S ) ) ) )
169, 13, 153sstr4d 3532 . 2  |-  ( ( J  e.  Top  /\  ( S  C_  X  /\  T  C_  S ) )  ->  ( ( int `  J ) `  T
)  C_  ( ( int `  J ) `  S ) )
17163impb 1190 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   U.cuni 4235   ` cfv 5570   Topctop 19564   intcnt 19688   clsccl 19689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19569  df-cld 19690  df-ntr 19691  df-cls 19692
This theorem is referenced by:  ntrin  19732  ntrcls0  19747  dvreslem  22482  dvres2lem  22483  dvaddbr  22510  dvmulbr  22511  dvcnvrelem2  22588  ntruni  30388  cldregopn  30392  limciccioolb  31869  limcicciooub  31885  cncfiooicclem1  31938
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