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Theorem isopn3 17132
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 17102 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
3 inss2 3564 . . . . . . . 8  |-  ( J  i^i  ~P S ) 
C_  ~P S
43unissi 4040 . . . . . . 7  |-  U. ( J  i^i  ~P S ) 
C_  U. ~P S
5 unipw 4416 . . . . . . 7  |-  U. ~P S  =  S
64, 5sseqtri 3382 . . . . . 6  |-  U. ( J  i^i  ~P S ) 
C_  S
76a1i 11 . . . . 5  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S ) 
C_  S )
8 id 21 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  J )
9 pwidg 3813 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  ~P S )
10 elin 3532 . . . . . . 7  |-  ( S  e.  ( J  i^i  ~P S )  <->  ( S  e.  J  /\  S  e. 
~P S ) )
118, 9, 10sylanbrc 647 . . . . . 6  |-  ( S  e.  J  ->  S  e.  ( J  i^i  ~P S ) )
12 elssuni 4045 . . . . . 6  |-  ( S  e.  ( J  i^i  ~P S )  ->  S  C_ 
U. ( J  i^i  ~P S ) )
1311, 12syl 16 . . . . 5  |-  ( S  e.  J  ->  S  C_ 
U. ( J  i^i  ~P S ) )
147, 13eqssd 3367 . . . 4  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S )  =  S )
152, 14sylan9eq 2490 . . 3  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  S  e.  J
)  ->  ( ( int `  J ) `  S )  =  S )
1615ex 425 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  ->  ( ( int `  J
) `  S )  =  S ) )
171ntropn 17115 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
18 eleq1 2498 . . 3  |-  ( ( ( int `  J
) `  S )  =  S  ->  ( ( ( int `  J
) `  S )  e.  J  <->  S  e.  J
) )
1917, 18syl5ibcom 213 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  =  S  ->  S  e.  J )
)
2016, 19impbid 185 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   ` cfv 5456   Topctop 16960   intcnt 17083
This theorem is referenced by:  ntridm  17134  ntrtop  17136  ntr0  17147  isopn3i  17148  opnnei  17186  cnntr  17341  llycmpkgen2  17584  dvnres  19819  dvcnvre  19905  taylthlem2  20292  ulmdvlem3  20320  abelth  20359  opnbnd  26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-ntr 17086
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