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Theorem ntrdif 18783
Description: An interior of a complement is the complement of the closure. This set is also known as the exterior of  A. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrdif  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  A )
) )

Proof of Theorem ntrdif
StepHypRef Expression
1 difss 3586 . . . 4  |-  ( X 
\  A )  C_  X
2 clscld.1 . . . . 5  |-  X  = 
U. J
32ntrval2 18782 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  A ) 
C_  X )  -> 
( ( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
41, 3mpan2 671 . . 3  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
54adantr 465 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
6 simpr 461 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  X )
7 dfss4 3687 . . . . 5  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
86, 7sylib 196 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( X  \  ( X  \  A ) )  =  A )
98fveq2d 5798 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
109difeq2d 3577 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) )  =  ( X  \ 
( ( cls `  J
) `  A )
) )
115, 10eqtrd 2493 1  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3428    C_ wss 3431   U.cuni 4194   ` cfv 5521   Topctop 18625   intcnt 18748   clsccl 18749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-top 18630  df-cld 18750  df-ntr 18751  df-cls 18752
This theorem is referenced by:  clsun  28666
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