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Theorem clsfval 17091
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 16981 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4385 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5967 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |-> 
|^| { y  e.  (
Clsd `  J )  |  x  C_  y } )  e.  _V )
52, 3, 43syl 19 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |-> 
|^| { y  e.  (
Clsd `  J )  |  x  C_  y } )  e.  _V )
6 unieq 4026 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2488 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3806 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 fveq2 5730 . . . . . 6  |-  ( j  =  J  ->  ( Clsd `  j )  =  ( Clsd `  J
) )
10 rabeq 2952 . . . . . 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 4057 . . . 4  |-  ( j  =  J  ->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y }  =  |^| { y  e.  ( Clsd `  J )  |  x 
C_  y } )
138, 12mpteq12dv 4289 . . 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 17087 . . 3  |-  cls  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  |^| { y  e.  ( Clsd `  j
)  |  x  C_  y } ) )
1513, 14fvmptg 5806 . 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 651 1  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( x  e.  ~P X  |->  |^| { y  e.  ( Clsd `  J
)  |  x  C_  y } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   |^|cint 4052    e. cmpt 4268   ` cfv 5456   Topctop 16960   Clsdccld 17082   clsccl 17084
This theorem is referenced by:  clsval  17103  clsf  17114  mrccls  17145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-cls 17087
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