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Theorem kur14lem2 29517
Description: Lemma for kur14 29526. 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 19846 . . 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 5852 . 2  |-  ( I `
 A )  =  ( ( int `  J
) `  A )
8 kur14lem.k . . . 4  |-  K  =  ( cls `  J
)
98fveq1i 5852 . . 3  |-  ( K `
 ( X  \  A ) )  =  ( ( cls `  J
) `  ( X  \  A ) )
109difeq2i 3560 . 2  |-  ( X 
\  ( K `  ( X  \  A ) ) )  =  ( X  \  ( ( cls `  J ) `
 ( X  \  A ) ) )
115, 7, 103eqtr4i 2443 1  |-  ( I `
 A )  =  ( X  \  ( K `  ( X  \  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    e. wcel 1844    \ cdif 3413    C_ wss 3416   U.cuni 4193   ` cfv 5571   Topctop 19688   intcnt 19812   clsccl 19813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-top 19693  df-cld 19814  df-ntr 19815  df-cls 19816
This theorem is referenced by:  kur14lem6  29521  kur14lem7  29522
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