MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isopn3 Structured version   Unicode version

Theorem isopn3 19433
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 19403 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
3 inss2 3701 . . . . . . . 8  |-  ( J  i^i  ~P S ) 
C_  ~P S
43unissi 4253 . . . . . . 7  |-  U. ( J  i^i  ~P S ) 
C_  U. ~P S
5 unipw 4683 . . . . . . 7  |-  U. ~P S  =  S
64, 5sseqtri 3518 . . . . . 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 4006 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  ~P S )
108, 9elind 3670 . . . . . 6  |-  ( S  e.  J  ->  S  e.  ( J  i^i  ~P S ) )
11 elssuni 4260 . . . . . 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 3503 . . . 4  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S )  =  S )
142, 13sylan9eq 2502 . . 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 19416 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
17 eleq1 2513 . . 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 1381    e. wcel 1802    i^i cin 3457    C_ wss 3458   ~Pcpw 3993   U.cuni 4230   ` cfv 5574   Topctop 19261   intcnt 19384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-top 19266  df-ntr 19387
This theorem is referenced by:  ntridm  19435  ntrtop  19437  ntr0  19448  isopn3i  19449  opnnei  19487  cnntr  19642  llycmpkgen2  19917  dvnres  22200  dvcnvre  22286  taylthlem2  22634  ulmdvlem3  22662  abelth  22701  opnbnd  30111  ioontr  31481  cncfuni  31592  fperdvper  31615  dirkercncflem3  31772  dirkercncflem4  31773  fourierdlem58  31832  fourierdlem73  31847
  Copyright terms: Public domain W3C validator