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Theorem ntridm 17056
Description: The interior operation is idempotent. (Contributed by NM, 2-Oct-2007.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntridm  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  ( ( int `  J ) `  S ) )  =  ( ( int `  J
) `  S )
)

Proof of Theorem ntridm
StepHypRef Expression
1 clscld.1 . . 3  |-  X  = 
U. J
21ntropn 17037 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
31ntrss3 17048 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  X )
41isopn3 17054 . . 3  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  S )  C_  X )  ->  (
( ( int `  J
) `  S )  e.  J  <->  ( ( int `  J ) `  (
( int `  J
) `  S )
)  =  ( ( int `  J ) `
 S ) ) )
53, 4syldan 457 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  e.  J  <->  ( ( int `  J ) `  ( ( int `  J
) `  S )
)  =  ( ( int `  J ) `
 S ) ) )
62, 5mpbid 202 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  ( ( int `  J ) `  S ) )  =  ( ( int `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264   U.cuni 3958   ` cfv 5395   Topctop 16882   intcnt 17005
This theorem is referenced by:  dvmptntr  19725  cldregopn  26026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-top 16887  df-ntr 17008
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