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

Theorem cldregopn 26352
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 16800 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  e.  ( Clsd `  J
) )
3 eqcom 2298 . . . . 5  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A  <->  A  =  (
( int `  J
) `  ( ( cls `  J ) `  A ) ) )
43biimpi 186 . . . 4  |-  ( ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A  ->  A  =  ( ( int `  J
) `  ( ( cls `  J ) `  A ) ) )
5 fveq2 5541 . . . . . 6  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( ( int `  J ) `  c )  =  ( ( int `  J
) `  ( ( cls `  J ) `  A ) ) )
65eqeq2d 2307 . . . . 5  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( A  =  ( ( int `  J ) `  c
)  <->  A  =  (
( int `  J
) `  ( ( cls `  J ) `  A ) ) ) )
76rspcev 2897 . . . 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 463 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A )  ->  E. c  e.  ( Clsd `  J
) A  =  ( ( int `  J
) `  c )
)
98ex 423 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( int `  J ) `  (
( cls `  J
) `  A )
)  =  A  ->  E. c  e.  ( Clsd `  J ) A  =  ( ( int `  J ) `  c
) ) )
10 cldrcl 16779 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  J  e.  Top )
111cldss 16782 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  c  C_  X )
121ntrss2 16810 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  c  C_  X )  -> 
( ( int `  J
) `  c )  C_  c )
1310, 11, 12syl2anc 642 . . . . . . . 8  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  c
)
141clsss2 16825 . . . . . . . 8  |-  ( ( c  e.  ( Clsd `  J )  /\  (
( int `  J
) `  c )  C_  c )  ->  (
( cls `  J
) `  ( ( int `  J ) `  c ) )  C_  c )
1513, 14mpdan 649 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  c )
)  C_  c )
161ntrss 16808 . . . . . . 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 1182 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) ) 
C_  ( ( int `  J ) `  c
) )
181ntridm 16821 . . . . . . . 8  |-  ( ( J  e.  Top  /\  c  C_  X )  -> 
( ( int `  J
) `  ( ( int `  J ) `  c ) )  =  ( ( int `  J
) `  c )
)
1910, 11, 18syl2anc 642 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( int `  J
) `  c )
)  =  ( ( int `  J ) `
 c ) )
201ntrss3 16813 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  c  C_  X )  -> 
( ( int `  J
) `  c )  C_  X )
2110, 11, 20syl2anc 642 . . . . . . . . 9  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  X
)
221clsss3 16812 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  c )  C_  X )  ->  (
( cls `  J
) `  ( ( int `  J ) `  c ) )  C_  X )
2310, 21, 22syl2anc 642 . . . . . . . 8  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( cls `  J ) `  ( ( int `  J
) `  c )
)  C_  X )
241sscls 16809 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  c )  C_  X )  ->  (
( int `  J
) `  c )  C_  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )
2510, 21, 24syl2anc 642 . . . . . . . 8  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  (
( cls `  J
) `  ( ( int `  J ) `  c ) ) )
261ntrss 16808 . . . . . . . 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 1182 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( int `  J
) `  c )
)  C_  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) ) )
2819, 27eqsstr3d 3226 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  (
( int `  J
) `  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) ) )
2917, 28eqssd 3209 . . . . 5  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )  =  ( ( int `  J ) `  c
) )
3029adantl 452 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  c  e.  ( Clsd `  J ) )  ->  ( ( int `  J ) `  (
( cls `  J
) `  ( ( int `  J ) `  c ) ) )  =  ( ( int `  J ) `  c
) )
31 fveq2 5541 . . . . . 6  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( ( cls `  J ) `  A )  =  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )
3231fveq2d 5545 . . . . 5  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  A )
)  =  ( ( int `  J ) `
 ( ( cls `  J ) `  (
( int `  J
) `  c )
) ) )
33 id 19 . . . . 5  |-  ( A  =  ( ( int `  J ) `  c
)  ->  A  =  ( ( int `  J
) `  c )
)
3432, 33eqeq12d 2310 . . . 4  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( (
( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A  <->  ( ( int `  J ) `  (
( cls `  J
) `  ( ( int `  J ) `  c ) ) )  =  ( ( int `  J ) `  c
) ) )
3530, 34syl5ibrcom 213 . . 3  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  c  e.  ( Clsd `  J ) )  ->  ( A  =  ( ( int `  J
) `  c )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A ) )
3635rexlimdva 2680 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( E. c  e.  ( Clsd `  J
) A  =  ( ( int `  J
) `  c )  ->  ( ( int `  J
) `  ( ( cls `  J ) `  A ) )  =  A ) )
379, 36impbid 183 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   intcnt 16770   clsccl 16771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774
  Copyright terms: Public domain W3C validator