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

Theorem clsval 19705
Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
clsval  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
Distinct variable groups:    x, J    x, S    x, X

Proof of Theorem clsval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21clsfval 19693 . . . 4  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( y  e.  ~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) )
32fveq1d 5850 . . 3  |-  ( J  e.  Top  ->  (
( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
43adantr 463 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
51topopn 19582 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4600 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 16 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 483 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
91topcld 19703 . . . . 5  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
10 sseq2 3511 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1110rspcev 3207 . . . . 5  |-  ( ( X  e.  ( Clsd `  J )  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J
) S  C_  x
)
129, 11sylan 469 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J ) S 
C_  x )
13 intexrab 4596 . . . 4  |-  ( E. x  e.  ( Clsd `  J ) S  C_  x 
<-> 
|^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
1412, 13sylib 196 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  |^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
15 sseq1 3510 . . . . . 6  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
1615rabbidv 3098 . . . . 5  |-  ( y  =  S  ->  { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  {
x  e.  ( Clsd `  J )  |  S  C_  x } )
1716inteqd 4276 . . . 4  |-  ( y  =  S  ->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  |^| { x  e.  ( Clsd `  J )  |  S  C_  x } )
18 eqid 2454 . . . 4  |-  ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
)  =  ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
)
1917, 18fvmptg 5929 . . 3  |-  ( ( S  e.  ~P X  /\  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x }  e.  _V )  ->  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
208, 14, 19syl2anc 659 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( y  e. 
~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) `  S
)  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
214, 20eqtrd 2495 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   |^|cint 4271    |-> cmpt 4497   ` cfv 5570   Topctop 19561   Clsdccld 19684   clsccl 19686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19566  df-cld 19687  df-cls 19689
This theorem is referenced by:  cldcls  19710  clscld  19715  clsf  19716  clsval2  19718  clsss  19722  sscls  19724
  Copyright terms: Public domain W3C validator