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Theorem topbnd 25610
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 16753 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  A )
) )
32ineq2d 3345 . . 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 3387 . . 3  |-  ( ( ( cls `  J
) `  A )  i^i  ( X  \  (
( int `  J
) `  A )
) )  =  ( ( ( ( cls `  J ) `  A
)  i^i  X )  \  ( ( int `  J ) `  A
) )
53, 4syl6eq 2306 . 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 16759 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  C_  X )
7 df-ss 3141 . . . 4  |-  ( ( ( cls `  J
) `  A )  C_  X  <->  ( ( ( cls `  J ) `
 A )  i^i 
X )  =  ( ( cls `  J
) `  A )
)
86, 7sylib 190 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( cls `  J ) `  A
)  i^i  X )  =  ( ( cls `  J ) `  A
) )
98difeq1d 3268 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( ( ( cls `  J ) `
 A )  i^i 
X )  \  (
( int `  J
) `  A )
)  =  ( ( ( cls `  J
) `  A )  \  ( ( int `  J ) `  A
) ) )
105, 9eqtrd 2290 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 6    /\ wa 360    = wceq 1619    e. wcel 1621    \ cdif 3124    i^i cin 3126    C_ wss 3127   U.cuni 3801   ` cfv 4673   Topctop 16594   intcnt 16717   clsccl 16718
This theorem is referenced by:  opnbnd  25611
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-top 16599  df-cld 16719  df-ntr 16720  df-cls 16721
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