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Theorem cldregopn 30076
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 19416 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  e.  ( Clsd `  J
) )
3 eqcom 2476 . . . . 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 5872 . . . . . 6  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( ( int `  J ) `  c )  =  ( ( int `  J
) `  ( ( cls `  J ) `  A ) ) )
65eqeq2d 2481 . . . . 5  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( A  =  ( ( int `  J ) `  c
)  <->  A  =  (
( int `  J
) `  ( ( cls `  J ) `  A ) ) ) )
76rspcev 3219 . . . 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 19395 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  J  e.  Top )
111cldss 19398 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  c  C_  X )
121ntrss2 19426 . . . . . . . . 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 19441 . . . . . . . 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 19424 . . . . . . 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 1228 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) ) 
C_  ( ( int `  J ) `  c
) )
181ntridm 19437 . . . . . . . 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 19429 . . . . . . . . . 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 19428 . . . . . . . . 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 19425 . . . . . . . . 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 19424 . . . . . . . 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 1228 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  ( ( int `  J
) `  c )
)  C_  ( ( int `  J ) `  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) ) )
2819, 27eqsstr3d 3544 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  (
( int `  J
) `  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) ) )
2917, 28eqssd 3526 . . . . 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 5872 . . . . . 6  |-  ( A  =  ( ( int `  J ) `  c
)  ->  ( ( cls `  J ) `  A )  =  ( ( cls `  J
) `  ( ( int `  J ) `  c ) ) )
3231fveq2d 5876 . . . . 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 2489 . . . 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 2959 . 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 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   U.cuni 4251   ` cfv 5594   Topctop 19263   Clsdccld 19385   intcnt 19386   clsccl 19387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19268  df-cld 19388  df-ntr 19389  df-cls 19390
This theorem is referenced by: (None)
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