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Theorem isopn3 12303
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 12288 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
3 inss2 3297 . . . . . . . 8  |-  ( J  i^i  ~P S ) 
C_  ~P S
43unissi 3759 . . . . . . 7  |-  U. ( J  i^i  ~P S ) 
C_  U. ~P S
5 unipw 4139 . . . . . . 7  |-  U. ~P S  =  S
64, 5sseqtri 3131 . . . . . 6  |-  U. ( J  i^i  ~P S ) 
C_  S
76a1i 9 . . . . 5  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S ) 
C_  S )
8 id 19 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  J )
9 pwidg 3524 . . . . . . 7  |-  ( S  e.  J  ->  S  e.  ~P S )
108, 9elind 3261 . . . . . 6  |-  ( S  e.  J  ->  S  e.  ( J  i^i  ~P S ) )
11 elssuni 3764 . . . . . 6  |-  ( S  e.  ( J  i^i  ~P S )  ->  S  C_ 
U. ( J  i^i  ~P S ) )
1210, 11syl 14 . . . . 5  |-  ( S  e.  J  ->  S  C_ 
U. ( J  i^i  ~P S ) )
137, 12eqssd 3114 . . . 4  |-  ( S  e.  J  ->  U. ( J  i^i  ~P S )  =  S )
142, 13sylan9eq 2192 . . 3  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  S  e.  J
)  ->  ( ( int `  J ) `  S )  =  S )
1514ex 114 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  ->  ( ( int `  J
) `  S )  =  S ) )
161ntropn 12295 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  e.  J )
17 eleq1 2202 . . 3  |-  ( ( ( int `  J
) `  S )  =  S  ->  ( ( ( int `  J
) `  S )  e.  J  <->  S  e.  J
) )
1816, 17syl5ibcom 154 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( int `  J ) `  S
)  =  S  ->  S  e.  J )
)
1915, 18impbid 128 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  J  <->  ( ( int `  J
) `  S )  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480    i^i cin 3070    C_ wss 3071   ~Pcpw 3510   U.cuni 3736   ` cfv 5123   Topctop 12173   intcnt 12271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12174  df-ntr 12274
This theorem is referenced by:  ntridm  12304  ntrtop  12306  ntr0  12312  isopn3i  12313  cnntr  12403
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