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Theorem opnregcld 26025
Description: A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
Hypothesis
Ref Expression
opnregcld.1  |-  X  = 
U. J
Assertion
Ref Expression
opnregcld  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
) )
Distinct variable groups:    A, o    o, J    o, X

Proof of Theorem opnregcld
StepHypRef Expression
1 opnregcld.1 . . . . 5  |-  X  = 
U. J
21ntropn 17037 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
3 eqcom 2390 . . . . 5  |-  ( ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) )
43biimpi 187 . . . 4  |-  ( ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  ->  A  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
5 fveq2 5669 . . . . . 6  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( ( cls `  J ) `  o )  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
65eqeq2d 2399 . . . . 5  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( A  =  ( ( cls `  J ) `  o
)  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) ) )
76rspcev 2996 . . . 4  |-  ( ( ( ( int `  J
) `  A )  e.  J  /\  A  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )  ->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) )
82, 4, 7syl2an 464 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A )  ->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
)
98ex 424 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  ->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) ) )
10 simpl 444 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  J  e.  Top )
111eltopss 16904 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  X )
121clsss3 17047 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
( ( cls `  J
) `  o )  C_  X )
1311, 12syldan 457 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  X )
141ntrss2 17045 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  ->  (
( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  ( ( cls `  J
) `  o )
)
1513, 14syldan 457 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  ( ( cls `  J
) `  o )
)
161clsss 17042 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X  /\  ( ( int `  J ) `
 ( ( cls `  J ) `  o
) )  C_  (
( cls `  J
) `  o )
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) ) 
C_  ( ( cls `  J ) `  (
( cls `  J
) `  o )
) )
1710, 13, 15, 16syl3anc 1184 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  ( ( cls `  J ) `  o ) ) )
181clsidm 17055 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
( ( cls `  J
) `  ( ( cls `  J ) `  o ) )  =  ( ( cls `  J
) `  o )
)
1911, 18syldan 457 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( cls `  J ) `  o ) )  =  ( ( cls `  J
) `  o )
)
2017, 19sseqtrd 3328 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  o )
)
211ntrss3 17048 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  ->  (
( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X )
2213, 21syldan 457 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X )
23 simpr 448 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  e.  J )
241sscls 17044 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
o  C_  ( ( cls `  J ) `  o ) )
2511, 24syldan 457 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( cls `  J ) `  o ) )
261ssntr 17046 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  /\  (
o  e.  J  /\  o  C_  ( ( cls `  J ) `  o
) ) )  -> 
o  C_  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )
2710, 13, 23, 25, 26syl22anc 1185 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )
281clsss 17042 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X  /\  o  C_  (
( int `  J
) `  ( ( cls `  J ) `  o ) ) )  ->  ( ( cls `  J ) `  o
)  C_  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) ) )
2910, 22, 27, 28syl3anc 1184 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) ) )
3020, 29eqssd 3309 . . . . 5  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  =  ( ( cls `  J
) `  o )
)
3130adantlr 696 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  o  e.  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )  =  ( ( cls `  J ) `  o
) )
32 fveq2 5669 . . . . . 6  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( int `  J ) `  A )  =  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )
3332fveq2d 5673 . . . . 5  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  A )
)  =  ( ( cls `  J ) `
 ( ( int `  J ) `  (
( cls `  J
) `  o )
) ) )
34 id 20 . . . . 5  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  A  =  ( ( cls `  J
) `  o )
)
3533, 34eqeq12d 2402 . . . 4  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( (
( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  <->  ( ( cls `  J ) `  (
( int `  J
) `  ( ( cls `  J ) `  o ) ) )  =  ( ( cls `  J ) `  o
) ) )
3631, 35syl5ibrcom 214 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  o  e.  J
)  ->  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  A )
)  =  A ) )
3736rexlimdva 2774 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( E. o  e.  J  A  =  ( ( cls `  J
) `  o )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A ) )
389, 37impbid 184 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  <->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651    C_ wss 3264   U.cuni 3958   ` cfv 5395   Topctop 16882   intcnt 17005   clsccl 17006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-top 16887  df-cld 17007  df-ntr 17008  df-cls 17009
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