MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clsfval Structured version   Unicode version

Theorem clsfval 19392
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 19282 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4617 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 6123 . . 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 4238 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2500 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3998 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5852 . . . . . 6  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
10 rabeq 3087 . . . . . 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 4272 . . . 4  |-  ( j  =  J  ->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y }  =  |^| { y  e.  ( Clsd `  J )  |  x 
C_  y } )
138, 12mpteq12dv 4511 . . 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 19388 . . 3  |-  cls  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } ) )
1513, 14fvmptg 5935 . 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 1381    e. wcel 1802   {crab 2795   _Vcvv 3093    C_ wss 3458   ~Pcpw 3993   U.cuni 4230   |^|cint 4267    |-> cmpt 4491   ` cfv 5574   Topctop 19261   Clsdccld 19383   clsccl 19385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-top 19266  df-cls 19388
This theorem is referenced by:  clsval  19404  clsf  19415  mrccls  19446
  Copyright terms: Public domain W3C validator