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Theorem topbnd 25653
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 16786 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  A )
) )
32ineq2d 3371 . . 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 3413 . . 3  |-  ( ( ( cls `  J
) `  A )  i^i  ( X  \  (
( int `  J
) `  A )
) )  =  ( ( ( ( cls `  J ) `  A
)  i^i  X )  \  ( ( int `  J ) `  A
) )
53, 4syl6eq 2332 . 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 16792 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  C_  X )
7 df-ss 3167 . . . 4  |-  ( ( ( cls `  J
) `  A )  C_  X  <->  ( ( ( cls `  J ) `
 A )  i^i 
X )  =  ( ( cls `  J
) `  A )
)
86, 7sylib 188 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  X )  =  ( ( cls `  J ) `  A
) )
98difeq1d 3294 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i 
X )  \  (
( int `  J
) `  A )
)  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
105, 9eqtrd 2316 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 358    = wceq 1623    e. wcel 1685    \ cdif 3150    i^i cin 3152    C_ wss 3153   U.cuni 3828   ` cfv 5221   Topctop 16627   intcnt 16750   clsccl 16751
This theorem is referenced by:  opnbnd  25654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-top 16632  df-cld 16752  df-ntr 16753  df-cls 16754
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