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Theorem topbnd 26217
Description: Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
Hypothesis
Ref Expression
topbnd.1  |-  X  = 
U. J
Assertion
Ref Expression
topbnd  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )

Proof of Theorem topbnd
StepHypRef Expression
1 topbnd.1 . . . . 5  |-  X  = 
U. J
21clsdif 17072 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  A )
) )
32ineq2d 3502 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  i^i  ( X  \  (
( int `  J
) `  A )
) ) )
4 indif2 3544 . . 3  |-  ( ( ( cls `  J
) `  A )  i^i  ( X  \  (
( int `  J
) `  A )
) )  =  ( ( ( ( cls `  J ) `  A
)  i^i  X )  \  ( ( int `  J ) `  A
) )
53, 4syl6eq 2452 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( ( cls `  J ) `  A
)  i^i  X )  \  ( ( int `  J ) `  A
) ) )
61clsss3 17078 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  C_  X )
7 df-ss 3294 . . . 4  |-  ( ( ( cls `  J
) `  A )  C_  X  <->  ( ( ( cls `  J ) `
 A )  i^i 
X )  =  ( ( cls `  J
) `  A )
)
86, 7sylib 189 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  X )  =  ( ( cls `  J ) `  A
) )
98difeq1d 3424 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i 
X )  \  (
( int `  J
) `  A )
)  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
105, 9eqtrd 2436 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  ( ( cls `  J ) `  ( X  \  A ) ) )  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3277    i^i cin 3279    C_ wss 3280   U.cuni 3975   ` cfv 5413   Topctop 16913   intcnt 17036   clsccl 17037
This theorem is referenced by:  opnbnd  26218
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-top 16918  df-cld 17038  df-ntr 17039  df-cls 17040
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