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Theorem clsdif 17118
Description: A closure of a complement is the complement of the interior. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
clsdif  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  A )
) )

Proof of Theorem clsdif
StepHypRef Expression
1 difss 3475 . . . 4  |-  ( X 
\  A )  C_  X
2 clscld.1 . . . . 5  |-  X  = 
U. J
32clsval2 17115 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  A ) 
C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
41, 3mpan2 654 . . 3  |-  ( J  e.  Top  ->  (
( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
54adantr 453 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
6 simpr 449 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  X )
7 dfss4 3576 . . . . 5  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
86, 7sylib 190 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( X  \  ( X  \  A ) )  =  A )
98fveq2d 5733 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( int `  J
) `  A )
)
109difeq2d 3466 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( X  \  (
( int `  J
) `  ( X  \  ( X  \  A
) ) ) )  =  ( X  \ 
( ( int `  J
) `  A )
) )
115, 10eqtrd 2469 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  A ) )  =  ( X  \  (
( int `  J
) `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3318    C_ wss 3321   U.cuni 4016   ` cfv 5455   Topctop 16959   intcnt 17082   clsccl 17083
This theorem is referenced by:  maxlp  17212  topbnd  26328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-top 16964  df-cld 17084  df-ntr 17085  df-cls 17086
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