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Theorem fclsval 20487
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 457 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  J  e.  Top )
2 fvssunirn 5879 . . . . 5  |-  ( Fil `  Y )  C_  U. ran  Fil
32sseli 3485 . . . 4  |-  ( F  e.  ( Fil `  Y
)  ->  F  e.  U.
ran  Fil )
43adantl 466 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  e.  U. ran  Fil )
5 filn0 20341 . . . . . 6  |-  ( F  e.  ( Fil `  Y
)  ->  F  =/=  (/) )
65adantl 466 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  =/=  (/) )
7 fvex 5866 . . . . . 6  |-  ( ( cls `  J ) `
 t )  e. 
_V
87rgenw 2804 . . . . 5  |-  A. t  e.  F  ( ( cls `  J ) `  t )  e.  _V
9 iinexg 4597 . . . . 5  |-  ( ( F  =/=  (/)  /\  A. t  e.  F  (
( cls `  J
) `  t )  e.  _V )  ->  |^|_ t  e.  F  ( ( cls `  J ) `  t )  e.  _V )
106, 8, 9sylancl 662 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  |^|_ t  e.  F  ( ( cls `  J
) `  t )  e.  _V )
11 0ex 4567 . . . 4  |-  (/)  e.  _V
12 ifcl 3968 . . . 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 662 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J ) `  t
) ,  (/) )  e. 
_V )
14 unieq 4242 . . . . . . 7  |-  ( j  =  J  ->  U. j  =  U. J )
15 fclsval.x . . . . . . 7  |-  X  = 
U. J
1614, 15syl6eqr 2502 . . . . . 6  |-  ( j  =  J  ->  U. j  =  X )
17 unieq 4242 . . . . . 6  |-  ( f  =  F  ->  U. f  =  U. F )
1816, 17eqeqan12d 2466 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( U. j  = 
U. f  <->  X  =  U. F ) )
19 iineq1 4330 . . . . . . 7  |-  ( f  =  F  ->  |^|_ t  e.  f  ( ( cls `  j ) `  t )  =  |^|_ t  e.  F  (
( cls `  j
) `  t )
)
2019adantl 466 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  j
) `  t )
)
21 simpll 753 . . . . . . . . 9  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  j  =  J )
2221fveq2d 5860 . . . . . . . 8  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  ( cls `  j )  =  ( cls `  J
) )
2322fveq1d 5858 . . . . . . 7  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  (
( cls `  j
) `  t )  =  ( ( cls `  J ) `  t
) )
2423iineq2dv 4338 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  F  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2520, 24eqtrd 2484 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2618, 25ifbieq1d 3949 . . . 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
) ,  (/) ) )
27 df-fcls 20420 . . . 4  |-  fClus  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ t  e.  f  (
( cls `  j
) `  t ) ,  (/) ) )
2826, 27ovmpt2ga 6417 . . 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 ) ,  (/) ) )
291, 4, 13, 28syl3anc 1229 . 2  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) ) )
30 filunibas 20360 . . . . 5  |-  ( F  e.  ( Fil `  Y
)  ->  U. F  =  Y )
3130eqeq2d 2457 . . . 4  |-  ( F  e.  ( Fil `  Y
)  ->  ( X  =  U. F  <->  X  =  Y ) )
3231adantl 466 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( X  =  U. F 
<->  X  =  Y ) )
3332ifbid 3948 . 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 ) ,  (/) ) )
3429, 33eqtrd 2484 1  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  Y ,  |^|_ t  e.  F  (
( cls `  J
) `  t ) ,  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095   (/)c0 3770   ifcif 3926   U.cuni 4234   |^|_ciin 4316   ran crn 4990   ` cfv 5578  (class class class)co 6281   Topctop 19372   clsccl 19497   Filcfil 20324    fClus cfcls 20415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-int 4272  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-fbas 18395  df-fil 20325  df-fcls 20420
This theorem is referenced by:  isfcls  20488  fclscmpi  20508
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