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Theorem ntrdif 19638
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 3545 . . . 4  |-  ( X 
\  A )  C_  X
2 clscld.1 . . . . 5  |-  X  = 
U. J
32ntrval2 19637 . . . 4  |-  ( ( J  e.  Top  /\  ( X  \  A ) 
C_  X )  -> 
( ( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
41, 3mpan2 669 . . 3  |-  ( J  e.  Top  ->  (
( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
54adantr 463 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( X  \  A ) )  =  ( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) ) )
6 simpr 459 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  ->  A  C_  X )
7 dfss4 3657 . . . . 5  |-  ( A 
C_  X  <->  ( X  \  ( X  \  A
) )  =  A )
86, 7sylib 196 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( X  \  ( X  \  A ) )  =  A )
98fveq2d 5778 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( X  \  ( X  \  A
) ) )  =  ( ( cls `  J
) `  A )
)
109difeq2d 3536 . 2  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( X  \  (
( cls `  J
) `  ( X  \  ( X  \  A
) ) ) )  =  ( X  \ 
( ( cls `  J
) `  A )
) )
115, 10eqtrd 2423 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 367    = wceq 1399    e. wcel 1826    \ cdif 3386    C_ wss 3389   U.cuni 4163   ` cfv 5496   Topctop 19479   intcnt 19603   clsccl 19604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-top 19484  df-cld 19605  df-ntr 19606  df-cls 19607
This theorem is referenced by:  clsun  30312
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