Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  opnregcld Structured version   Unicode version

Theorem opnregcld 26324
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 17105 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
3 eqcom 2437 . . . . 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 5720 . . . . . 6  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( ( cls `  J ) `  o )  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
65eqeq2d 2446 . . . . 5  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( A  =  ( ( cls `  J ) `  o
)  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) ) )
76rspcev 3044 . . . 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 16972 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  X )
121clsss3 17115 . . . . . . . . 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 17113 . . . . . . . . 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 17110 . . . . . . . 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 17123 . . . . . . . 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 3376 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  o )
)
211ntrss3 17116 . . . . . . . 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 17112 . . . . . . . . 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 17114 . . . . . . . 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 17110 . . . . . . 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 3357 . . . . 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 5720 . . . . . 6  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( int `  J ) `  A )  =  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )
3332fveq2d 5724 . . . . 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 2449 . . . 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 2822 . 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 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   U.cuni 4007   ` cfv 5446   Topctop 16950   intcnt 17073   clsccl 17074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-cld 17075  df-ntr 17076  df-cls 17077
  Copyright terms: Public domain W3C validator