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Theorem isopn3 18689
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 18659 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
3 inss2 3590 . . . . . . . 8  |-  ( J  i^i  ~P S ) 
C_  ~P S
43unissi 4133 . . . . . . 7  |-  U. ( J  i^i  ~P S ) 
C_  U. ~P S
5 unipw 4561 . . . . . . 7  |-  U. ~P S  =  S
64, 5sseqtri 3407 . . . . . 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 22 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  J )
9 pwidg 3892 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  ~P S )
108, 9elind 3559 . . . . . 6  |-  ( S  e.  J  ->  S  e.  ( J  i^i  ~P S ) )
11 elssuni 4140 . . . . . 6  |-  ( S  e.  ( J  i^i  ~P S )  ->  S  C_ 
U. ( J  i^i  ~P S ) )
1210, 11syl 16 . . . . 5  |-  ( S  e.  J  ->  S  C_ 
U. ( J  i^i  ~P S ) )
137, 12eqssd 3392 . . . 4  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S )  =  S )
142, 13sylan9eq 2495 . . 3  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  S  e.  J
)  ->  ( ( int `  J ) `  S )  =  S )
1514ex 434 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  ->  ( ( int `  J
) `  S )  =  S ) )
161ntropn 18672 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
17 eleq1 2503 . . 3  |-  ( ( ( int `  J
) `  S )  =  S  ->  ( ( ( int `  J
) `  S )  e.  J  <->  S  e.  J
) )
1816, 17syl5ibcom 220 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  =  S  ->  S  e.  J )
)
1915, 18impbid 191 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( ( int `  J
) `  S )  =  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3346    C_ wss 3347   ~Pcpw 3879   U.cuni 4110   ` cfv 5437   Topctop 18517   intcnt 18640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-top 18522  df-ntr 18643
This theorem is referenced by:  ntridm  18691  ntrtop  18693  ntr0  18704  isopn3i  18705  opnnei  18743  cnntr  18898  llycmpkgen2  19142  dvnres  21424  dvcnvre  21510  taylthlem2  21858  ulmdvlem3  21886  abelth  21925  opnbnd  28543
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