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Theorem clsval 12312
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 12302 . . . 4  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( y  e.  ~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) )
32fveq1d 5429 . . 3  |-  ( J  e.  Top  ->  (
( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
43adantr 274 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
5 eqid 2140 . . 3  |-  ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
)  =  ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
)
6 sseq1 3123 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
76rabbidv 2678 . . . 4  |-  ( y  =  S  ->  { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  {
x  e.  ( Clsd `  J )  |  S  C_  x } )
87inteqd 3782 . . 3  |-  ( y  =  S  ->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  |^| { x  e.  ( Clsd `  J )  |  S  C_  x } )
91topopn 12207 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
10 elpw2g 4087 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
119, 10syl 14 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
1211biimpar 295 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
131topcld 12310 . . . . 5  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
14 sseq2 3124 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1514rspcev 2792 . . . . 5  |-  ( ( X  e.  ( Clsd `  J )  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J
) S  C_  x
)
1613, 15sylan 281 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J ) S 
C_  x )
17 intexrabim 4084 . . . 4  |-  ( E. x  e.  ( Clsd `  J ) S  C_  x  ->  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x }  e.  _V )
1816, 17syl 14 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  |^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
195, 8, 12, 18fvmptd3 5520 . 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 } )
204, 19eqtrd 2173 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   E.wrex 2418   {crab 2421   _Vcvv 2689    C_ wss 3074   ~Pcpw 3513   U.cuni 3742   |^|cint 3777    |-> cmpt 3995   ` cfv 5129   Topctop 12196   Clsdccld 12293   clsccl 12295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-top 12197  df-cld 12296  df-cls 12298
This theorem is referenced by:  cldcls  12315  clsss  12319  sscls  12321
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