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Theorem opnregcld 28537
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 18665 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
3 eqcom 2445 . . . . 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 5703 . . . . . 6  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( ( cls `  J ) `  o )  =  ( ( cls `  J
) `  ( ( int `  J ) `  A ) ) )
65eqeq2d 2454 . . . . 5  |-  ( o  =  ( ( int `  J ) `  A
)  ->  ( A  =  ( ( cls `  J ) `  o
)  <->  A  =  (
( cls `  J
) `  ( ( int `  J ) `  A ) ) ) )
76rspcev 3085 . . . 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 18532 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  X )
121clsss3 18675 . . . . . . . . 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 18673 . . . . . . . . 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 18670 . . . . . . . 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 1218 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  ( ( cls `  J ) `  o ) ) )
181clsidm 18683 . . . . . . . 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 3404 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )  C_  (
( cls `  J
) `  o )
)
211ntrss3 18676 . . . . . . . 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 18672 . . . . . . . . 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 18674 . . . . . . . 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 1219 . . . . . . 7  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  o  C_  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) )
281clsss 18670 . . . . . . 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 1218 . . . . . 6  |-  ( ( J  e.  Top  /\  o  e.  J )  ->  ( ( cls `  J
) `  o )  C_  ( ( cls `  J
) `  ( ( int `  J ) `  ( ( cls `  J
) `  o )
) ) )
3020, 29eqssd 3385 . . . . 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 5703 . . . . . 6  |-  ( A  =  ( ( cls `  J ) `  o
)  ->  ( ( int `  J ) `  A )  =  ( ( int `  J
) `  ( ( cls `  J ) `  o ) ) )
3332fveq2d 5707 . . . . 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 2457 . . . 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 2853 . 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 1369    e. wcel 1756   E.wrex 2728    C_ wss 3340   U.cuni 4103   ` cfv 5430   Topctop 18510   intcnt 18633   clsccl 18634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-top 18515  df-cld 18635  df-ntr 18636  df-cls 18637
This theorem is referenced by: (None)
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