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Theorem fclsval 20272
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 5889 . . . . 5  |-  ( Fil `  Y )  C_  U. ran  Fil
32sseli 3500 . . . 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 20126 . . . . . 6  |-  ( F  e.  ( Fil `  Y
)  ->  F  =/=  (/) )
65adantl 466 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  =/=  (/) )
7 fvex 5876 . . . . . 6  |-  ( ( cls `  J ) `
 t )  e. 
_V
87rgenw 2825 . . . . 5  |-  A. t  e.  F  ( ( cls `  J ) `  t )  e.  _V
9 iinexg 4607 . . . . 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 4577 . . . 4  |-  (/)  e.  _V
12 ifcl 3981 . . . 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 4253 . . . . . . 7  |-  ( j  =  J  ->  U. j  =  U. J )
15 fclsval.x . . . . . . 7  |-  X  = 
U. J
1614, 15syl6eqr 2526 . . . . . 6  |-  ( j  =  J  ->  U. j  =  X )
17 unieq 4253 . . . . . 6  |-  ( f  =  F  ->  U. f  =  U. F )
1816, 17eqeqan12d 2490 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( U. j  = 
U. f  <->  X  =  U. F ) )
19 iineq1 4340 . . . . . . 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 5870 . . . . . . . 8  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  ( cls `  j )  =  ( cls `  J
) )
2322fveq1d 5868 . . . . . . 7  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  (
( cls `  j
) `  t )  =  ( ( cls `  J ) `  t
) )
2423iineq2dv 4348 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  F  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2520, 24eqtrd 2508 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2618, 25ifbieq1d 3962 . . . 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 20205 . . . 4  |-  fClus  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ t  e.  f  (
( cls `  j
) `  t ) ,  (/) ) )
2826, 27ovmpt2ga 6416 . . 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 1228 . 2  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) ) )
30 filunibas 20145 . . . . 5  |-  ( F  e.  ( Fil `  Y
)  ->  U. F  =  Y )
3130eqeq2d 2481 . . . 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 3961 . 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 2508 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   (/)c0 3785   ifcif 3939   U.cuni 4245   |^|_ciin 4326   ran crn 5000   ` cfv 5588  (class class class)co 6284   Topctop 19189   clsccl 19313   Filcfil 20109    fClus cfcls 20200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-fbas 18215  df-fil 20110  df-fcls 20205
This theorem is referenced by:  isfcls  20273  fclscmpi  20293
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