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Theorem kur14lem5 24888
Description: Lemma for kur14 24894. Closure is an idempotent operation in the set of subsets of a topology. (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
kur14lem5  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)

Proof of Theorem kur14lem5
StepHypRef Expression
1 kur14lem.j . . 3  |-  J  e. 
Top
2 kur14lem.a . . 3  |-  A  C_  X
3 kur14lem.x . . . 4  |-  X  = 
U. J
43clsidm 17123 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  ( ( cls `  J ) `  A ) )  =  ( ( cls `  J
) `  A )
)
51, 2, 4mp2an 654 . 2  |-  ( ( cls `  J ) `
 ( ( cls `  J ) `  A
) )  =  ( ( cls `  J
) `  A )
6 kur14lem.k . . 3  |-  K  =  ( cls `  J
)
76fveq1i 5721 . . 3  |-  ( K `
 A )  =  ( ( cls `  J
) `  A )
86, 7fveq12i 5725 . 2  |-  ( K `
 ( K `  A ) )  =  ( ( cls `  J
) `  ( ( cls `  J ) `  A ) )
95, 8, 73eqtr4i 2465 1  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    C_ wss 3312   U.cuni 4007   ` cfv 5446   Topctop 16950   intcnt 17073   clsccl 17074
This theorem is referenced by:  kur14lem6  24889  kur14lem7  24890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-cld 17075  df-cls 17077
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