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Theorem ntridm 16805
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 16786 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
31ntrss3 16797 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  X )
41isopn3 16803 . . 3  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  S )  C_  X )  ->  (
( ( int `  J
) `  S )  e.  J  <->  ( ( int `  J ) `  (
( int `  J
) `  S )
)  =  ( ( int `  J ) `
 S ) ) )
53, 4syldan 456 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  e.  J  <->  ( ( int `  J ) `  ( ( int `  J
) `  S )
)  =  ( ( int `  J ) `
 S ) ) )
62, 5mpbid 201 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   ` cfv 5255   Topctop 16631   intcnt 16754
This theorem is referenced by:  dvmptntr  19320  cldregopn  26249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-ntr 16757
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