Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  kur14lem5 Unicode version

Theorem kur14lem5 24676
Description: Lemma for kur14 24682. 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 17055 . . 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 5670 . . 3  |-  ( K `
 A )  =  ( ( cls `  J
) `  A )
86, 7fveq12i 5674 . 2  |-  ( K `
 ( K `  A ) )  =  ( ( cls `  J
) `  ( ( cls `  J ) `  A ) )
95, 8, 73eqtr4i 2418 1  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    C_ wss 3264   U.cuni 3958   ` cfv 5395   Topctop 16882   intcnt 17005   clsccl 17006
This theorem is referenced by:  kur14lem6  24677  kur14lem7  24678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-top 16887  df-cld 17007  df-cls 17009
  Copyright terms: Public domain W3C validator