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Theorem fclsval 17703
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 443 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  J  e.  Top )
2 fvssunirn 5551 . . . . 5  |-  ( Fil `  Y )  C_  U. ran  Fil
32sseli 3176 . . . 4  |-  ( F  e.  ( Fil `  Y
)  ->  F  e.  U.
ran  Fil )
43adantl 452 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  e.  U. ran  Fil )
5 filn0 17557 . . . . . 6  |-  ( F  e.  ( Fil `  Y
)  ->  F  =/=  (/) )
65adantl 452 . . . . 5  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  F  =/=  (/) )
7 fvex 5539 . . . . . 6  |-  ( ( cls `  J ) `
 t )  e. 
_V
87rgenw 2610 . . . . 5  |-  A. t  e.  F  ( ( cls `  J ) `  t )  e.  _V
9 iinexg 4171 . . . . 5  |-  ( ( F  =/=  (/)  /\  A. t  e.  F  (
( cls `  J
) `  t )  e.  _V )  ->  |^|_ t  e.  F  ( ( cls `  J ) `  t )  e.  _V )
106, 8, 9sylancl 643 . . . 4  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  |^|_ t  e.  F  ( ( cls `  J
) `  t )  e.  _V )
11 0ex 4150 . . . 4  |-  (/)  e.  _V
12 ifcl 3601 . . . 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 643 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  ->  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J ) `  t
) ,  (/) )  e. 
_V )
14 unieq 3836 . . . . . . 7  |-  ( j  =  J  ->  U. j  =  U. J )
15 fclsval.x . . . . . . 7  |-  X  = 
U. J
1614, 15syl6eqr 2333 . . . . . 6  |-  ( j  =  J  ->  U. j  =  X )
17 unieq 3836 . . . . . 6  |-  ( f  =  F  ->  U. f  =  U. F )
1816, 17eqeqan12d 2298 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  ->  ( U. j  = 
U. f  <->  X  =  U. F ) )
19 iineq1 3919 . . . . . . 7  |-  ( f  =  F  ->  |^|_ t  e.  f  ( ( cls `  j ) `  t )  =  |^|_ t  e.  F  (
( cls `  j
) `  t )
)
2019adantl 452 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  j
) `  t )
)
21 simpll 730 . . . . . . . . 9  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  j  =  J )
2221fveq2d 5529 . . . . . . . 8  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  ( cls `  j )  =  ( cls `  J
) )
2322fveq1d 5527 . . . . . . 7  |-  ( ( ( j  =  J  /\  f  =  F )  /\  t  e.  F )  ->  (
( cls `  j
) `  t )  =  ( ( cls `  J ) `  t
) )
2423iineq2dv 3927 . . . . . 6  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  F  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
2520, 24eqtrd 2315 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
|^|_ t  e.  f  ( ( cls `  j
) `  t )  =  |^|_ t  e.  F  ( ( cls `  J
) `  t )
)
26 eqidd 2284 . . . . 5  |-  ( ( j  =  J  /\  f  =  F )  -> 
(/)  =  (/) )
2718, 25, 26ifbieq12d 3587 . . . 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 17636 . . . 4  |-  fClus  =  ( j  e.  Top , 
f  e.  U. ran  Fil  |->  if ( U. j  =  U. f ,  |^|_ t  e.  f  (
( cls `  j
) `  t ) ,  (/) ) )
2927, 28ovmpt2ga 5977 . . 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 1182 . 2  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( J  fClus  F )  =  if ( X  =  U. F ,  |^|_ t  e.  F  ( ( cls `  J
) `  t ) ,  (/) ) )
31 filunibas 17576 . . . . 5  |-  ( F  e.  ( Fil `  Y
)  ->  U. F  =  Y )
3231eqeq2d 2294 . . . 4  |-  ( F  e.  ( Fil `  Y
)  ->  ( X  =  U. F  <->  X  =  Y ) )
3332adantl 452 . . 3  |-  ( ( J  e.  Top  /\  F  e.  ( Fil `  Y ) )  -> 
( X  =  U. F 
<->  X  =  Y ) )
3433ifbid 3583 . 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 2315 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788   (/)c0 3455   ifcif 3565   U.cuni 3827   |^|_ciin 3906   ran crn 4690   ` cfv 5255  (class class class)co 5858   Topctop 16631   clsccl 16755   Filcfil 17540    fClus cfcls 17631
This theorem is referenced by:  isfcls  17704  fclscmpi  17724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fbas 17520  df-fil 17541  df-fcls 17636
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