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Theorem cldbnd 29749
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 19359 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  A )  =  A ) )
3 eqimss 3556 . . . 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 3726 . . 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 3724 . . . . . 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 3691 . . . . . 6  |-  ( ( X  \  A )  i^i  A )  =  ( A  i^i  ( X  \  A ) )
10 disjdif 3899 . . . . . 6  |-  ( A  i^i  ( X  \  A ) )  =  (/)
119, 10eqtri 2496 . . . . 5  |-  ( ( X  \  A )  i^i  A )  =  (/)
12 sseq0 3817 . . . . 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 3691 . . . . . . . 8  |-  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `
 ( X  \  A ) )  i^i  ( ( cls `  J
) `  A )
)
16 dfss4 3732 . . . . . . . . . . 11  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
17 fveq2 5866 . . . . . . . . . . . 12  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
1817eqcomd 2475 . . . . . . . . . . 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 3700 . . . . . . . 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 2520 . . . . . . 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 3700 . . . . . 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 2469 . . . . 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 3631 . . . . . . 7  |-  ( X 
\  A )  C_  X
261opnbnd 29748 . . . . . . 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 19328 . . . . . . 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 2539 . . . . . . 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 1379    e. wcel 1767    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   U.cuni 4245   ` cfv 5588   Topctop 19189   Clsdccld 19311   clsccl 19313
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-top 19194  df-cld 19314  df-ntr 19315  df-cls 19316
This theorem is referenced by: (None)
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