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Theorem cldbnd 28526
Description: A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
Hypothesis
Ref Expression
opnbnd.1  |-  X  = 
U. J
Assertion
Ref Expression
cldbnd  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A ) )

Proof of Theorem cldbnd
StepHypRef Expression
1 opnbnd.1 . . . . 5  |-  X  = 
U. J
21iscld3 18673 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  A )  =  A ) )
3 eqimss 3413 . . . 4  |-  ( ( ( cls `  J
) `  A )  =  A  ->  ( ( cls `  J ) `
 A )  C_  A )
42, 3syl6bi 228 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( cls `  J
) `  A )  C_  A ) )
5 ssinss1 3583 . . 3  |-  ( ( ( cls `  J
) `  A )  C_  A  ->  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A )
64, 5syl6 33 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  C_  A
) )
7 sslin 3581 . . . . . 6  |-  ( ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A  ->  (
( X  \  A
)  i^i  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) )  C_  ( ( X  \  A )  i^i 
A ) )
87adantl 466 . . . . 5  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  C_  A
)  ->  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  C_  ( ( X  \  A )  i^i  A
) )
9 incom 3548 . . . . . 6  |-  ( ( X  \  A )  i^i  A )  =  ( A  i^i  ( X  \  A ) )
10 disjdif 3756 . . . . . 6  |-  ( A  i^i  ( X  \  A ) )  =  (/)
119, 10eqtri 2463 . . . . 5  |-  ( ( X  \  A )  i^i  A )  =  (/)
12 sseq0 3674 . . . . 5  |-  ( ( ( ( X  \  A )  i^i  (
( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) )  C_  ( ( X  \  A )  i^i 
A )  /\  (
( X  \  A
)  i^i  A )  =  (/) )  ->  (
( X  \  A
)  i^i  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) )  =  (/) )
138, 11, 12sylancl 662 . . . 4  |-  ( ( ( J  e.  Top  /\  A  C_  X )  /\  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  C_  A
)  ->  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/) )
1413ex 434 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A  ->  (
( X  \  A
)  i^i  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) )  =  (/) ) )
15 incom 3548 . . . . . . . 8  |-  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `
 ( X  \  A ) )  i^i  ( ( cls `  J
) `  A )
)
16 dfss4 3589 . . . . . . . . . . 11  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
17 fveq2 5696 . . . . . . . . . . . 12  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
1817eqcomd 2448 . . . . . . . . . . 11  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
1916, 18sylbi 195 . . . . . . . . . 10  |-  ( A 
C_  X  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2019adantl 466 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2120ineq2d 3557 . . . . . . . 8  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  ( X  \  A ) )  i^i  ( ( cls `  J ) `  A
) )  =  ( ( ( cls `  J
) `  ( X  \  A ) )  i^i  ( ( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
2215, 21syl5eq 2487 . . . . . . 7  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  ( X  \  A ) )  i^i  ( ( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
2322ineq2d 3557 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  i^i  (
( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) )  =  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `
 ( X  \  A ) )  i^i  ( ( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) ) )
2423eqeq1d 2451 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( X 
\  A )  i^i  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/) 
<->  ( ( X  \  A )  i^i  (
( ( cls `  J
) `  ( X  \  A ) )  i^i  ( ( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )  =  (/) ) )
25 difss 3488 . . . . . . 7  |-  ( X 
\  A )  C_  X
261opnbnd 28525 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  A ) 
C_  X )  -> 
( ( X  \  A )  e.  J  <->  ( ( X  \  A
)  i^i  ( (
( cls `  J
) `  ( X  \  A ) )  i^i  ( ( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )  =  (/) ) )
2725, 26mpan2 671 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  <->  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `  ( X  \  A ) )  i^i  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) ) )  =  (/) ) )
2827adantr 465 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  <->  ( ( X  \  A
)  i^i  ( (
( cls `  J
) `  ( X  \  A ) )  i^i  ( ( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )  =  (/) ) )
2924, 28bitr4d 256 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( X 
\  A )  i^i  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/) 
<->  ( X  \  A
)  e.  J ) )
301opncld 18642 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  A )  e.  J )  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) )
3130ex 434 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
3231adantr 465 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  ( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
33 eleq1 2503 . . . . . . 7  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3416, 33sylbi 195 . . . . . 6  |-  ( A 
C_  X  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3534adantl 466 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \ 
( X  \  A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
3632, 35sylibd 214 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  A  e.  ( Clsd `  J ) ) )
3729, 36sylbid 215 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( X 
\  A )  i^i  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) ) )  =  (/)  ->  A  e.  (
Clsd `  J )
) )
3814, 37syld 44 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A  ->  A  e.  ( Clsd `  J
) ) )
396, 38impbid 191 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642   U.cuni 4096   ` cfv 5423   Topctop 18503   Clsdccld 18625   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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