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Theorem isopn3 16819
Description: A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
isopn3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( ( int `  J
) `  S )  =  S ) )

Proof of Theorem isopn3
StepHypRef Expression
1 clscld.1 . . . . 5  |-  X  = 
U. J
21ntrval 16789 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
3 inss2 3403 . . . . . . . 8  |-  ( J  i^i  ~P S ) 
C_  ~P S
4 uniss 3864 . . . . . . . 8  |-  ( ( J  i^i  ~P S
)  C_  ~P S  ->  U. ( J  i^i  ~P S )  C_  U. ~P S )
53, 4ax-mp 8 . . . . . . 7  |-  U. ( J  i^i  ~P S ) 
C_  U. ~P S
6 unipw 4240 . . . . . . 7  |-  U. ~P S  =  S
75, 6sseqtri 3223 . . . . . 6  |-  U. ( J  i^i  ~P S ) 
C_  S
87a1i 10 . . . . 5  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S ) 
C_  S )
9 id 19 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  J )
10 pwidg 3650 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  ~P S )
11 elin 3371 . . . . . . 7  |-  ( S  e.  ( J  i^i  ~P S )  <->  ( S  e.  J  /\  S  e. 
~P S ) )
129, 10, 11sylanbrc 645 . . . . . 6  |-  ( S  e.  J  ->  S  e.  ( J  i^i  ~P S ) )
13 elssuni 3871 . . . . . 6  |-  ( S  e.  ( J  i^i  ~P S )  ->  S  C_ 
U. ( J  i^i  ~P S ) )
1412, 13syl 15 . . . . 5  |-  ( S  e.  J  ->  S  C_ 
U. ( J  i^i  ~P S ) )
158, 14eqssd 3209 . . . 4  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S )  =  S )
162, 15sylan9eq 2348 . . 3  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  S  e.  J
)  ->  ( ( int `  J ) `  S )  =  S )
1716ex 423 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  ->  ( ( int `  J
) `  S )  =  S ) )
181ntropn 16802 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
19 eleq1 2356 . . 3  |-  ( ( ( int `  J
) `  S )  =  S  ->  ( ( ( int `  J
) `  S )  e.  J  <->  S  e.  J
) )
2018, 19syl5ibcom 211 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  =  S  ->  S  e.  J )
)
2117, 20impbid 183 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( ( int `  J
) `  S )  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   ` cfv 5271   Topctop 16647   intcnt 16770
This theorem is referenced by:  ntridm  16821  ntrtop  16823  ntr0  16834  isopn3i  16835  opnnei  16873  cnntr  17020  llycmpkgen2  17261  dvnres  19296  dvcnvre  19382  taylthlem2  19769  ulmdvlem3  19795  abelth  19833  opnbnd  26346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-ntr 16773
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