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Theorem ntridm 13286
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 13277 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
31ntrss3 13283 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  C_  X )
41isopn3 13285 . . 3  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  S )  C_  X )  ->  (
( ( int `  J
) `  S )  e.  J  <->  ( ( int `  J ) `  (
( int `  J
) `  S )
)  =  ( ( int `  J ) `
 S ) ) )
53, 4syldan 282 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  e.  J  <->  ( ( int `  J ) `  ( ( int `  J
) `  S )
)  =  ( ( int `  J ) `
 S ) ) )
62, 5mpbid 147 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    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    C_ wss 3129   U.cuni 3807   ` cfv 5212   Topctop 13155   intcnt 13253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-top 13156  df-ntr 13256
This theorem is referenced by: (None)
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