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Theorem cldbnd 26343
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 17133 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  A )  =  A ) )
3 eqimss 3402 . . . 4  |-  ( ( ( cls `  J
) `  A )  =  A  ->  ( ( cls `  J ) `
 A )  C_  A )
42, 3syl6bi 221 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( cls `  J
) `  A )  C_  A ) )
5 ssinss1 3571 . . 3  |-  ( ( ( cls `  J
) `  A )  C_  A  ->  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A )
64, 5syl6 32 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  C_  A
) )
7 sslin 3569 . . . . . 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 454 . . . . 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 3535 . . . . . 6  |-  ( ( X  \  A )  i^i  A )  =  ( A  i^i  ( X  \  A ) )
10 disjdif 3702 . . . . . 6  |-  ( A  i^i  ( X  \  A ) )  =  (/)
119, 10eqtri 2458 . . . . 5  |-  ( ( X  \  A )  i^i  A )  =  (/)
12 sseq0 3661 . . . . 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 645 . . . 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 425 . . 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 3535 . . . . . . . 8  |-  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `
 ( X  \  A ) )  i^i  ( ( cls `  J
) `  A )
)
16 dfss4 3577 . . . . . . . . . . 11  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
17 fveq2 5731 . . . . . . . . . . . 12  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
1817eqcomd 2443 . . . . . . . . . . 11  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
1916, 18sylbi 189 . . . . . . . . . 10  |-  ( A 
C_  X  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2019adantl 454 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2120ineq2d 3544 . . . . . . . 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 2482 . . . . . . 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 3544 . . . . . 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 2446 . . . . 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 3476 . . . . . . 7  |-  ( X 
\  A )  C_  X
261opnbnd 26342 . . . . . . 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 654 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  <->  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `  ( X  \  A ) )  i^i  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) ) )  =  (/) ) )
2827adantr 453 . . . . 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 249 . . . 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 17102 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  A )  e.  J )  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) )
3130ex 425 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
3231adantr 453 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  ( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
33 eleq1 2498 . . . . . . 7  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3416, 33sylbi 189 . . . . . 6  |-  ( A 
C_  X  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3534adantl 454 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \ 
( X  \  A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
3632, 35sylibd 207 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  A  e.  ( Clsd `  J ) ) )
3729, 36sylbid 208 . . 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 43 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A  ->  A  e.  ( Clsd `  J
) ) )
396, 38impbid 185 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   U.cuni 4017   ` cfv 5457   Topctop 16963   Clsdccld 17085   clsccl 17087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-top 16968  df-cld 17088  df-ntr 17089  df-cls 17090
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