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Theorem cldbnd 25836
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 17018 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  A )  =  A ) )
3 eqimss 3316 . . . 4  |-  ( ( ( cls `  J
) `  A )  =  A  ->  ( ( cls `  J ) `
 A )  C_  A )
42, 3syl6bi 219 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( cls `  J
) `  A )  C_  A ) )
5 ssinss1 3485 . . 3  |-  ( ( ( cls `  J
) `  A )  C_  A  ->  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A )
64, 5syl6 29 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  C_  A
) )
7 sslin 3483 . . . . . 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 452 . . . . 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 3449 . . . . . 6  |-  ( ( X  \  A )  i^i  A )  =  ( A  i^i  ( X  \  A ) )
10 disjdif 3615 . . . . . 6  |-  ( A  i^i  ( X  \  A ) )  =  (/)
119, 10eqtri 2386 . . . . 5  |-  ( ( X  \  A )  i^i  A )  =  (/)
12 sseq0 3574 . . . . 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 643 . . . 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 423 . . 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 3449 . . . . . . . 8  |-  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `
 ( X  \  A ) )  i^i  ( ( cls `  J
) `  A )
)
16 dfss4 3491 . . . . . . . . . . 11  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
17 fveq2 5632 . . . . . . . . . . . 12  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
1817eqcomd 2371 . . . . . . . . . . 11  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
1916, 18sylbi 187 . . . . . . . . . 10  |-  ( A 
C_  X  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2019adantl 452 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2120ineq2d 3458 . . . . . . . 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 2410 . . . . . . 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 3458 . . . . . 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 2374 . . . . 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 3390 . . . . . . 7  |-  ( X 
\  A )  C_  X
261opnbnd 25835 . . . . . . 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 652 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  <->  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `  ( X  \  A ) )  i^i  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) ) )  =  (/) ) )
2827adantr 451 . . . . 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 247 . . . 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 16987 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  A )  e.  J )  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) )
3130ex 423 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
3231adantr 451 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  ( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
33 eleq1 2426 . . . . . . 7  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3416, 33sylbi 187 . . . . . 6  |-  ( A 
C_  X  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3534adantl 452 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \ 
( X  \  A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
3632, 35sylibd 205 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  A  e.  ( Clsd `  J ) ) )
3729, 36sylbid 206 . . 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 40 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A  ->  A  e.  ( Clsd `  J
) ) )
396, 38impbid 183 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 176    /\ wa 358    = wceq 1647    e. wcel 1715    \ cdif 3235    i^i cin 3237    C_ wss 3238   (/)c0 3543   U.cuni 3929   ` cfv 5358   Topctop 16848   Clsdccld 16970   clsccl 16972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-top 16853  df-cld 16973  df-ntr 16974  df-cls 16975
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