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Theorem fclsval 18040
Description: The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Hypothesis
Ref Expression
fclsval.x  |-  X  = 
U. J
Assertion
Ref Expression
fclsval  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  Y ,  |^|_ t  e.  F  (
( cls `  J
) `  t ) ,  (/) ) )
Distinct variable groups:    t, F    t, J
Allowed substitution hints:    X( t)    Y( t)

Proof of Theorem fclsval
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  J  e.  Top )
2 fvssunirn 5754 . . . . 5  |-  ( Fil `  Y )  C_  U. ran  Fil
32sseli 3344 . . . 4  |-  ( F  e.  ( Fil `  Y
)  ->  F  e.  U.
ran  Fil )
43adantl 453 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  e.  U. ran  Fil )
5 filn0 17894 . . . . . 6  |-  ( F  e.  ( Fil `  Y
)  ->  F  =/=  (/) )
65adantl 453 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  =/=  (/) )
7 fvex 5742 . . . . . 6  |-  ( ( cls `  J ) `
 t )  e. 
_V
87rgenw 2773 . . . . 5  |-  A. t  e.  F  ( ( cls `  J ) `  t )  e.  _V
9 iinexg 4360 . . . . 5  |-  ( ( F  =/=  (/)  /\  A. t  e.  F  (
( cls `  J
) `  t )  e.  _V )  ->  |^|_ t  e.  F  ( ( cls `  J ) `  t )  e.  _V )
106, 8, 9sylancl 644 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  |^|_ t  e.  F  ( ( cls `  J
) `  t )  e.  _V )
11 0ex 4339 . . . 4  |-  (/)  e.  _V
12 ifcl 3775 . . . 4  |-  ( (
|^|_ t  e.  F  ( ( cls `  J
) `  t )  e.  _V  /\  (/)  e.  _V )  ->  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) )  e.  _V )
1310, 11, 12sylancl 644 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J ) `  t
) ,  (/) )  e. 
_V )
14 unieq 4024 . . . . . . 7  |-  ( j  =  J  ->  U. j  =  U. J )
15 fclsval.x . . . . . . 7  |-  X  = 
U. J
1614, 15syl6eqr 2486 . . . . . 6  |-  ( j  =  J  ->  U. j  =  X )
17 unieq 4024 . . . . . 6  |-  ( f  =  F  ->  U. f  =  U. F )
1816, 17eqeqan12d 2451 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( U. j  = 
U. f  <->  X  =  U. F ) )
19 iineq1 4107 . . . . . . 7  |-  ( f  =  F  ->  |^|_ t  e.  f  ( ( cls `  j ) `  t )  =  |^|_ t  e.  F  (
( cls `  j
) `  t )
)
2019adantl 453 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  j
) `  t )
)
21 simpll 731 . . . . . . . . 9  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  j  =  J )
2221fveq2d 5732 . . . . . . . 8  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  ( cls `  j )  =  ( cls `  J
) )
2322fveq1d 5730 . . . . . . 7  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  (
( cls `  j
) `  t )  =  ( ( cls `  J ) `  t
) )
2423iineq2dv 4115 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  F  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2520, 24eqtrd 2468 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
26 eqidd 2437 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
(/)  =  (/) )
2718, 25, 26ifbieq12d 3761 . . . 4  |-  ( ( j  =  J  /\  f  =  F )  ->  if ( U. j  =  U. f ,  |^|_ t  e.  f  (
( cls `  j
) `  t ) ,  (/) )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J ) `  t
) ,  (/) ) )
28 df-fcls 17973 . . . 4  |-  fClus  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ t  e.  f  (
( cls `  j
) `  t ) ,  (/) ) )
2927, 28ovmpt2ga 6203 . . 3  |-  ( ( J  e.  Top  /\  F  e.  U. ran  Fil  /\  if ( X  = 
U. F ,  |^|_ t  e.  F  (
( cls `  J
) `  t ) ,  (/) )  e.  _V )  ->  ( J  fClus  F )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) ) )
301, 4, 13, 29syl3anc 1184 . 2  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) ) )
31 filunibas 17913 . . . . 5  |-  ( F  e.  ( Fil `  Y
)  ->  U. F  =  Y )
3231eqeq2d 2447 . . . 4  |-  ( F  e.  ( Fil `  Y
)  ->  ( X  =  U. F  <->  X  =  Y ) )
3332adantl 453 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( X  =  U. F 
<->  X  =  Y ) )
3433ifbid 3757 . 2  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J ) `  t
) ,  (/) )  =  if ( X  =  Y ,  |^|_ t  e.  F  ( ( cls `  J ) `  t ) ,  (/) ) )
3530, 34eqtrd 2468 1  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  Y ,  |^|_ t  e.  F  (
( cls `  J
) `  t ) ,  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956   (/)c0 3628   ifcif 3739   U.cuni 4015   |^|_ciin 4094   ran crn 4879   ` cfv 5454  (class class class)co 6081   Topctop 16958   clsccl 17082   Filcfil 17877    fClus cfcls 17968
This theorem is referenced by:  isfcls  18041  fclscmpi  18061
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-fbas 16699  df-fil 17878  df-fcls 17973
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