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Theorem kur14lem2 28291
Description: Lemma for kur14 28300. 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 19318 . . 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 5865 . 2  |-  ( I `
 A )  =  ( ( int `  J
) `  A )
8 kur14lem.k . . . 4  |-  K  =  ( cls `  J
)
98fveq1i 5865 . . 3  |-  ( K `
 ( X  \  A ) )  =  ( ( cls `  J
) `  ( X  \  A ) )
109difeq2i 3619 . 2  |-  ( X 
\  ( K `  ( X  \  A ) ) )  =  ( X  \  ( ( cls `  J ) `
 ( X  \  A ) ) )
115, 7, 103eqtr4i 2506 1  |-  ( I `
 A )  =  ( X  \  ( K `  ( X  \  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   U.cuni 4245   ` cfv 5586   Topctop 19161   intcnt 19284   clsccl 19285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-top 19166  df-cld 19286  df-ntr 19287  df-cls 19288
This theorem is referenced by:  kur14lem6  28295  kur14lem7  28296
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