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

Theorem clsfval 19371
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 19261 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4636 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 6140 . . 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 4258 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2526 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 4020 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5871 . . . . . 6  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
10 rabeq 3112 . . . . . 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 4292 . . . 4  |-  ( j  =  J  ->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y }  =  |^| { y  e.  ( Clsd `  J )  |  x 
C_  y } )
138, 12mpteq12dv 4530 . . 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 19367 . . 3  |-  cls  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } ) )
1513, 14fvmptg 5954 . 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 1379    e. wcel 1767   {crab 2821   _Vcvv 3118    C_ wss 3481   ~Pcpw 4015   U.cuni 4250   |^|cint 4287    |-> cmpt 4510   ` cfv 5593   Topctop 19240   Clsdccld 19362   clsccl 19364
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-top 19245  df-cls 19367
This theorem is referenced by:  clsval  19383  clsf  19394  mrccls  19425
  Copyright terms: Public domain W3C validator