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Theorem kur14lem2 27234
Description: Lemma for kur14 27243. Write interior in terms of closure and complement:  i A  =  c k c A where 
c is complement and  k is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j  |-  J  e. 
Top
kur14lem.x  |-  X  = 
U. J
kur14lem.k  |-  K  =  ( cls `  J
)
kur14lem.i  |-  I  =  ( int `  J
)
kur14lem.a  |-  A  C_  X
Assertion
Ref Expression
kur14lem2  |-  ( I `
 A )  =  ( X  \  ( K `  ( X  \  A ) ) )

Proof of Theorem kur14lem2
StepHypRef Expression
1 kur14lem.j . . 3  |-  J  e. 
Top
2 kur14lem.a . . 3  |-  A  C_  X
3 kur14lem.x . . . 4  |-  X  = 
U. J
43ntrval2 18782 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  =  ( X  \ 
( ( cls `  J
) `  ( X  \  A ) ) ) )
51, 2, 4mp2an 672 . 2  |-  ( ( int `  J ) `
 A )  =  ( X  \  (
( cls `  J
) `  ( X  \  A ) ) )
6 kur14lem.i . . 3  |-  I  =  ( int `  J
)
76fveq1i 5795 . 2  |-  ( I `
 A )  =  ( ( int `  J
) `  A )
8 kur14lem.k . . . 4  |-  K  =  ( cls `  J
)
98fveq1i 5795 . . 3  |-  ( K `
 ( X  \  A ) )  =  ( ( cls `  J
) `  ( X  \  A ) )
109difeq2i 3574 . 2  |-  ( X 
\  ( K `  ( X  \  A ) ) )  =  ( X  \  ( ( cls `  J ) `
 ( X  \  A ) ) )
115, 7, 103eqtr4i 2491 1  |-  ( I `
 A )  =  ( X  \  ( K `  ( X  \  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    = 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:  kur14lem6  27238  kur14lem7  27239
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