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Theorem clsfval 18634
Description: The closure function on the subsets of a topology's base set. (Contributed by NM, 3-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
cldval.1  |-  X  = 
U. J
Assertion
Ref Expression
clsfval  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( x  e.  ~P X  |->  |^| { y  e.  ( Clsd `  J
)  |  x  C_  y } ) )
Distinct variable groups:    x, y, J    x, X
Allowed substitution hint:    X( y)

Proof of Theorem clsfval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4  |-  X  = 
U. J
21topopn 18524 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4481 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5952 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |-> 
|^| { y  e.  (
Clsd `  J )  |  x  C_  y } )  e.  _V )
52, 3, 43syl 20 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |-> 
|^| { y  e.  (
Clsd `  J )  |  x  C_  y } )  e.  _V )
6 unieq 4104 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2493 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3870 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5696 . . . . . 6  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
10 rabeq 2971 . . . . . 6  |-  ( (
Clsd `  j )  =  ( Clsd `  J
)  ->  { y  e.  ( Clsd `  j
)  |  x  C_  y }  =  {
y  e.  ( Clsd `  J )  |  x 
C_  y } )
119, 10syl 16 . . . . 5  |-  ( j  =  J  ->  { y  e.  ( Clsd `  j
)  |  x  C_  y }  =  {
y  e.  ( Clsd `  J )  |  x 
C_  y } )
1211inteqd 4138 . . . 4  |-  ( j  =  J  ->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y }  =  |^| { y  e.  ( Clsd `  J )  |  x 
C_  y } )
138, 12mpteq12dv 4375 . . 3  |-  ( j  =  J  ->  (
x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } )  =  ( x  e.  ~P X  |-> 
|^| { y  e.  (
Clsd `  J )  |  x  C_  y } ) )
14 df-cls 18630 . . 3  |-  cls  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } ) )
1513, 14fvmptg 5777 . 2  |-  ( ( J  e.  Top  /\  ( x  e.  ~P X  |->  |^| { y  e.  ( Clsd `  J
)  |  x  C_  y } )  e.  _V )  ->  ( cls `  J
)  =  ( x  e.  ~P X  |->  |^|
{ y  e.  (
Clsd `  J )  |  x  C_  y } ) )
165, 15mpdan 668 1  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( x  e.  ~P X  |->  |^| { y  e.  ( Clsd `  J
)  |  x  C_  y } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   U.cuni 4096   |^|cint 4133    e. cmpt 4355   ` cfv 5423   Topctop 18503   Clsdccld 18625   clsccl 18627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-top 18508  df-cls 18630
This theorem is referenced by:  clsval  18646  clsf  18657  mrccls  18688
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