ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  clsval Unicode version

Theorem clsval 14279
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 14269 . . . 4  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( y  e.  ~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) )
32fveq1d 5556 . . 3  |-  ( J  e.  Top  ->  (
( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
43adantr 276 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
5 eqid 2193 . . 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 3202 . . . . 5  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
76rabbidv 2749 . . . 4  |-  ( y  =  S  ->  { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  {
x  e.  ( Clsd `  J )  |  S  C_  x } )
87inteqd 3875 . . 3  |-  ( y  =  S  ->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  |^| { x  e.  ( Clsd `  J )  |  S  C_  x } )
91topopn 14176 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
10 elpw2g 4185 . . . . 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 297 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
131topcld 14277 . . . . 5  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
14 sseq2 3203 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1514rspcev 2864 . . . . 5  |-  ( ( X  e.  ( Clsd `  J )  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J
) S  C_  x
)
1613, 15sylan 283 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J ) S 
C_  x )
17 intexrabim 4182 . . . 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 5651 . 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 2226 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 104    <-> wb 105    = wceq 1364    e. wcel 2164   E.wrex 2473   {crab 2476   _Vcvv 2760    C_ wss 3153   ~Pcpw 3601   U.cuni 3835   |^|cint 3870    |-> cmpt 4090   ` cfv 5254   Topctop 14165   Clsdccld 14260   clsccl 14262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-top 14166  df-cld 14263  df-cls 14265
This theorem is referenced by:  cldcls  14282  clsss  14286  sscls  14288
  Copyright terms: Public domain W3C validator