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Theorem cldregopn 30354
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 19675 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  e.  ( Clsd `  J
) )
3 eqcom 2466 . . . . 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 2471 . . . . 5  |-  ( c  =  ( ( cls `  J ) `  A
)  ->  ( A  =  ( ( int `  J ) `  c
)  <->  A  =  (
( int `  J
) `  ( ( cls `  J ) `  A ) ) ) )
76rspcev 3210 . . . 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 19654 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  J  e.  Top )
111cldss 19657 . . . . . . 7  |-  ( c  e.  ( Clsd `  J
)  ->  c  C_  X )
121ntrss2 19685 . . . . . . . . 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 19700 . . . . . . . 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 19683 . . . . . . 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 19696 . . . . . . . 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 19688 . . . . . . . . . 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 19687 . . . . . . . . 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 19684 . . . . . . . . 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 19683 . . . . . . . 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 3534 . . . . . 6  |-  ( c  e.  ( Clsd `  J
)  ->  ( ( int `  J ) `  c )  C_  (
( int `  J
) `  ( ( cls `  J ) `  ( ( int `  J
) `  c )
) ) )
2917, 28eqssd 3516 . . . . 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 2479 . . . 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 2949 . 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 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   U.cuni 4251   ` cfv 5594   Topctop 19521   Clsdccld 19644   intcnt 19645   clsccl 19646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 19526  df-cld 19647  df-ntr 19648  df-cls 19649
This theorem is referenced by: (None)
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