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Theorem fclsval 19586
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 5718 . . . . 5  |-  ( Fil `  Y )  C_  U. ran  Fil
32sseli 3357 . . . 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 19440 . . . . . 6  |-  ( F  e.  ( Fil `  Y
)  ->  F  =/=  (/) )
65adantl 466 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  =/=  (/) )
7 fvex 5706 . . . . . 6  |-  ( ( cls `  J ) `
 t )  e. 
_V
87rgenw 2788 . . . . 5  |-  A. t  e.  F  ( ( cls `  J ) `  t )  e.  _V
9 iinexg 4457 . . . . 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 4427 . . . 4  |-  (/)  e.  _V
12 ifcl 3836 . . . 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 4104 . . . . . . 7  |-  ( j  =  J  ->  U. j  =  U. J )
15 fclsval.x . . . . . . 7  |-  X  = 
U. J
1614, 15syl6eqr 2493 . . . . . 6  |-  ( j  =  J  ->  U. j  =  X )
17 unieq 4104 . . . . . 6  |-  ( f  =  F  ->  U. f  =  U. F )
1816, 17eqeqan12d 2458 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( U. j  = 
U. f  <->  X  =  U. F ) )
19 iineq1 4190 . . . . . . 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 5700 . . . . . . . 8  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  ( cls `  j )  =  ( cls `  J
) )
2322fveq1d 5698 . . . . . . 7  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  (
( cls `  j
) `  t )  =  ( ( cls `  J ) `  t
) )
2423iineq2dv 4198 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  F  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2520, 24eqtrd 2475 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2618, 25ifbieq1d 3817 . . . 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 19519 . . . 4  |-  fClus  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ t  e.  f  (
( cls `  j
) `  t ) ,  (/) ) )
2826, 27ovmpt2ga 6225 . . 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 1218 . 2  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) ) )
30 filunibas 19459 . . . . 5  |-  ( F  e.  ( Fil `  Y
)  ->  U. F  =  Y )
3130eqeq2d 2454 . . . 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 3816 . 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 2475 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 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   _Vcvv 2977   (/)c0 3642   ifcif 3796   U.cuni 4096   |^|_ciin 4177   ran crn 4846   ` cfv 5423  (class class class)co 6096   Topctop 18503   clsccl 18627   Filcfil 19423    fClus cfcls 19514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-fbas 17819  df-fil 19424  df-fcls 19519
This theorem is referenced by:  isfcls  19587  fclscmpi  19607
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