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Theorem opnregcld 30116
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 19416 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
3 eqcom 2450 . . . . 5  |-  ( ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) )
43biimpi 194 . . . 4  |-  ( ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  ->  A  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
5 fveq2 5852 . . . . . 6  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( ( cls `  J ) `  o )  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
65eqeq2d 2455 . . . . 5  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( A  =  ( ( cls `  J ) `  o
)  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) ) )
76rspcev 3194 . . . 4  |-  ( ( ( ( int `  J
) `  A )  e.  J  /\  A  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )  ->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) )
82, 4, 7syl2an 477 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A )  ->  E. o  e.  J  A  =  ( ( cls `  J
) `  o )
)
98ex 434 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  (
( int `  J
) `  A )
)  =  A  ->  E. o  e.  J  A  =  ( ( cls `  J ) `  o ) ) )
10 simpl 457 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  J  e.  Top )
111eltopss 19283 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  X )
121clsss3 19426 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
( ( cls `  J
) `  o )  C_  X )
1311, 12syldan 470 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  X )
141ntrss2 19424 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  ->  (
( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  ( ( cls `  J
) `  o )
)
1513, 14syldan 470 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  ( ( cls `  J
) `  o )
)
161clsss 19421 . . . . . . . 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 1227 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  ( ( cls `  J ) `  o ) ) )
181clsidm 19434 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
( ( cls `  J
) `  ( ( cls `  J ) `  o ) )  =  ( ( cls `  J
) `  o )
)
1911, 18syldan 470 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( cls `  J ) `  o ) )  =  ( ( cls `  J
) `  o )
)
2017, 19sseqtrd 3522 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  o )
)
211ntrss3 19427 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  o )  C_  X )  ->  (
( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X )
2213, 21syldan 470 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  o ) )  C_  X )
23 simpr 461 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  e.  J )
241sscls 19423 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  C_  X )  -> 
o  C_  ( ( cls `  J ) `  o ) )
2511, 24syldan 470 . . . . . . . 8  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( cls `  J ) `  o ) )
261ssntr 19425 . . . . . . . 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 1228 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )
281clsss 19421 . . . . . . 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 1227 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) ) )
3020, 29eqssd 3503 . . . . 5  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  =  ( ( cls `  J
) `  o )
)
3130adantlr 714 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  o  e.  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )  =  ( ( cls `  J ) `  o
) )
32 fveq2 5852 . . . . . 6  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( int `  J ) `  A )  =  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )
3332fveq2d 5856 . . . . 5  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  A )
)  =  ( ( cls `  J ) `
 ( ( int `  J ) `  (
( cls `  J
) `  o )
) ) )
34 id 22 . . . . 5  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  A  =  ( ( cls `  J
) `  o )
)
3533, 34eqeq12d 2463 . . . 4  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( (
( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A  <->  ( ( cls `  J ) `  (
( int `  J
) `  ( ( cls `  J ) `  o ) ) )  =  ( ( cls `  J ) `  o
) ) )
3631, 35syl5ibrcom 222 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  o  e.  J
)  ->  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  A )
)  =  A ) )
3736rexlimdva 2933 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( E. o  e.  J  A  =  ( ( cls `  J
) `  o )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  A ) )  =  A ) )
389, 37impbid 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E.wrex 2792    C_ wss 3458   U.cuni 4230   ` cfv 5574   Topctop 19261   intcnt 19384   clsccl 19385
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-int 4268  df-iun 4313  df-iin 4314  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-cld 19386  df-ntr 19387  df-cls 19388
This theorem is referenced by: (None)
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