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Theorem cldregopn 28531
Description: A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
Hypothesis
Ref Expression
opnregcld.1  |-  X  = 
U. J
Assertion
Ref Expression
cldregopn  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  (
( cls `  J
) `  A )
)  =  A  <->  E. c  e.  ( Clsd `  J
) A  =  ( ( int `  J
) `  c )
) )
Distinct variable groups:    A, c    J, c    X, c

Proof of Theorem cldregopn
StepHypRef Expression
1 opnregcld.1 . . . . 5  |-  X  = 
U. J
21clscld 18656 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  e.  ( Clsd `  J
) )
3 eqcom 2445 . . . . 5  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A  <->  A  =  (
( int `  J
) `  ( ( cls `  J ) `  A ) ) )
43biimpi 194 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A  ->  A  =  ( ( int `  J
) `  ( ( cls `  J ) `  A ) ) )
5 fveq2 5696 . . . . . 6  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( ( int `  J ) `  c )  =  ( ( int `  J
) `  ( ( cls `  J ) `  A ) ) )
65eqeq2d 2454 . . . . 5  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( A  =  ( ( int `  J ) `  c
)  <->  A  =  (
( int `  J
) `  ( ( cls `  J ) `  A ) ) ) )
76rspcev 3078 . . . 4  |-  ( ( ( ( cls `  J
) `  A )  e.  ( Clsd `  J
)  /\  A  =  ( ( int `  J
) `  ( ( cls `  J ) `  A ) ) )  ->  E. c  e.  (
Clsd `  J ) A  =  ( ( int `  J ) `  c ) )
82, 4, 7syl2an 477 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A )  ->  E. c  e.  ( Clsd `  J
) A  =  ( ( int `  J
) `  c )
)
98ex 434 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  (
( cls `  J
) `  A )
)  =  A  ->  E. c  e.  ( Clsd `  J ) A  =  ( ( int `  J ) `  c
) ) )
10 cldrcl 18635 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  J  e.  Top )
111cldss 18638 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  c  C_  X )
121ntrss2 18666 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  c  C_  X )  -> 
( ( int `  J
) `  c )  C_  c )
1310, 11, 12syl2anc 661 . . . . . . . 8  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  c
)
141clsss2 18681 . . . . . . . 8  |-  ( ( c  e.  ( Clsd `  J )  /\  (
( int `  J
) `  c )  C_  c )  ->  (
( cls `  J
) `  ( ( int `  J ) `  c ) )  C_  c )
1513, 14mpdan 668 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  c )
)  C_  c )
161ntrss 18664 . . . . . . 7  |-  ( ( J  e.  Top  /\  c  C_  X  /\  (
( cls `  J
) `  ( ( int `  J ) `  c ) )  C_  c )  ->  (
( int `  J
) `  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) )  C_  (
( int `  J
) `  c )
)
1710, 11, 15, 16syl3anc 1218 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) ) 
C_  ( ( int `  J ) `  c
) )
181ntridm 18677 . . . . . . . 8  |-  ( ( J  e.  Top  /\  c  C_  X )  -> 
( ( int `  J
) `  ( ( int `  J ) `  c ) )  =  ( ( int `  J
) `  c )
)
1910, 11, 18syl2anc 661 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( int `  J
) `  c )
)  =  ( ( int `  J ) `
 c ) )
201ntrss3 18669 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  c  C_  X )  -> 
( ( int `  J
) `  c )  C_  X )
2110, 11, 20syl2anc 661 . . . . . . . . 9  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  X
)
221clsss3 18668 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  c )  C_  X )  ->  (
( cls `  J
) `  ( ( int `  J ) `  c ) )  C_  X )
2310, 21, 22syl2anc 661 . . . . . . . 8  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  c )
)  C_  X )
241sscls 18665 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  c )  C_  X )  ->  (
( int `  J
) `  c )  C_  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )
2510, 21, 24syl2anc 661 . . . . . . . 8  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  (
( cls `  J
) `  ( ( int `  J ) `  c ) ) )
261ntrss 18664 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( ( cls `  J
) `  ( ( int `  J ) `  c ) )  C_  X  /\  ( ( int `  J ) `  c
)  C_  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) )  ->  (
( int `  J
) `  ( ( int `  J ) `  c ) )  C_  ( ( int `  J
) `  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) ) )
2710, 23, 25, 26syl3anc 1218 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( int `  J
) `  c )
)  C_  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) ) )
2819, 27eqsstr3d 3396 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  (
( int `  J
) `  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) ) )
2917, 28eqssd 3378 . . . . 5  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )  =  ( ( int `  J ) `  c
) )
3029adantl 466 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  c  e.  ( Clsd `  J ) )  ->  ( ( int `  J ) `  (
( cls `  J
) `  ( ( int `  J ) `  c ) ) )  =  ( ( int `  J ) `  c
) )
31 fveq2 5696 . . . . . 6  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( ( cls `  J ) `  A )  =  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )
3231fveq2d 5700 . . . . 5  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  A )
)  =  ( ( int `  J ) `
 ( ( cls `  J ) `  (
( int `  J
) `  c )
) ) )
33 id 22 . . . . 5  |-  ( A  =  ( ( int `  J ) `  c
)  ->  A  =  ( ( int `  J
) `  c )
)
3432, 33eqeq12d 2457 . . . 4  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( (
( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A  <->  ( ( int `  J ) `  (
( cls `  J
) `  ( ( int `  J ) `  c ) ) )  =  ( ( int `  J ) `  c
) ) )
3530, 34syl5ibrcom 222 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  c  e.  ( Clsd `  J ) )  ->  ( A  =  ( ( int `  J
) `  c )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A ) )
3635rexlimdva 2846 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( E. c  e.  ( Clsd `  J
) A  =  ( ( int `  J
) `  c )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A ) )
379, 36impbid 191 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  (
( cls `  J
) `  A )
)  =  A  <->  E. c  e.  ( Clsd `  J
) A  =  ( ( int `  J
) `  c )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721    C_ wss 3333   U.cuni 4096   ` cfv 5423   Topctop 18503   Clsdccld 18625   intcnt 18626   clsccl 18627
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-top 18508  df-cld 18628  df-ntr 18629  df-cls 18630
This theorem is referenced by: (None)
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