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Theorem cldbnd 26223
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 17087 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  A )  =  A ) )
3 eqimss 3364 . . . 4  |-  ( ( ( cls `  J
) `  A )  =  A  ->  ( ( cls `  J ) `
 A )  C_  A )
42, 3syl6bi 220 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( cls `  J
) `  A )  C_  A ) )
5 ssinss1 3533 . . 3  |-  ( ( ( cls `  J
) `  A )  C_  A  ->  ( (
( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A )
64, 5syl6 31 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( A  e.  (
Clsd `  J )  ->  ( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  C_  A
) )
7 sslin 3531 . . . . . 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 453 . . . . 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 3497 . . . . . 6  |-  ( ( X  \  A )  i^i  A )  =  ( A  i^i  ( X  \  A ) )
10 disjdif 3664 . . . . . 6  |-  ( A  i^i  ( X  \  A ) )  =  (/)
119, 10eqtri 2428 . . . . 5  |-  ( ( X  \  A )  i^i  A )  =  (/)
12 sseq0 3623 . . . . 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 644 . . . 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 424 . . 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 3497 . . . . . . . 8  |-  ( ( ( cls `  J
) `  A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) )  =  ( ( ( cls `  J ) `
 ( X  \  A ) )  i^i  ( ( cls `  J
) `  A )
)
16 dfss4 3539 . . . . . . . . . . 11  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
17 fveq2 5691 . . . . . . . . . . . 12  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
1817eqcomd 2413 . . . . . . . . . . 11  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
1916, 18sylbi 188 . . . . . . . . . 10  |-  ( A 
C_  X  ->  (
( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2019adantl 453 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  =  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) )
2120ineq2d 3506 . . . . . . . 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 2452 . . . . . . 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 3506 . . . . . 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 2416 . . . . 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 3438 . . . . . . 7  |-  ( X 
\  A )  C_  X
261opnbnd 26222 . . . . . . 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 653 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  <->  ( ( X  \  A )  i^i  ( ( ( cls `  J ) `  ( X  \  A ) )  i^i  ( ( cls `  J ) `  ( X  \  ( X  \  A ) ) ) ) )  =  (/) ) )
2827adantr 452 . . . . 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 248 . . . 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 17056 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( X  \  A )  e.  J )  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) )
3130ex 424 . . . . . 6  |-  ( J  e.  Top  ->  (
( X  \  A
)  e.  J  -> 
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
3231adantr 452 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  ( X  \  ( X  \  A ) )  e.  ( Clsd `  J
) ) )
33 eleq1 2468 . . . . . . 7  |-  ( ( X  \  ( X 
\  A ) )  =  A  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3416, 33sylbi 188 . . . . . 6  |-  ( A 
C_  X  ->  (
( X  \  ( X  \  A ) )  e.  ( Clsd `  J
)  <->  A  e.  ( Clsd `  J ) ) )
3534adantl 453 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \ 
( X  \  A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
3632, 35sylibd 206 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( X  \  A )  e.  J  ->  A  e.  ( Clsd `  J ) ) )
3729, 36sylbid 207 . . 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 42 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i  ( ( cls `  J
) `  ( X  \  A ) ) ) 
C_  A  ->  A  e.  ( Clsd `  J
) ) )
396, 38impbid 184 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3281    i^i cin 3283    C_ wss 3284   (/)c0 3592   U.cuni 3979   ` cfv 5417   Topctop 16917   Clsdccld 17039   clsccl 17041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-top 16922  df-cld 17042  df-ntr 17043  df-cls 17044
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