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Theorem topbnd 28524
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 18662 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  A )
) )
32ineq2d 3557 . . 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 3598 . . 3  |-  ( ( ( cls `  J
) `  A )  i^i  ( X  \  (
( int `  J
) `  A )
) )  =  ( ( ( ( cls `  J ) `  A
)  i^i  X )  \  ( ( int `  J ) `  A
) )
53, 4syl6eq 2491 . 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 18668 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  C_  X )
7 df-ss 3347 . . . 4  |-  ( ( ( cls `  J
) `  A )  C_  X  <->  ( ( ( cls `  J ) `
 A )  i^i 
X )  =  ( ( cls `  J
) `  A )
)
86, 7sylib 196 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  X )  =  ( ( cls `  J ) `  A
) )
98difeq1d 3478 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i 
X )  \  (
( int `  J
) `  A )
)  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
105, 9eqtrd 2475 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3330    i^i cin 3332    C_ wss 3333   U.cuni 4096   ` cfv 5423   Topctop 18503   intcnt 18626   clsccl 18627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-top 18508  df-cld 18628  df-ntr 18629  df-cls 18630
This theorem is referenced by:  opnbnd  28525
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