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Theorem fclstopon 20245
Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
fclstopon  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )

Proof of Theorem fclstopon
StepHypRef Expression
1 fclstop 20244 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )
2 istopon 19190 . . . 4  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
32baib 901 . . 3  |-  ( J  e.  Top  ->  ( J  e.  (TopOn `  X
)  <->  X  =  U. J ) )
41, 3syl 16 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  X )  <->  X  =  U. J ) )
5 eqid 2467 . . . . 5  |-  U. J  =  U. J
65fclsfil 20243 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  F  e.  ( Fil `  U. J
) )
7 fveq2 5864 . . . . 5  |-  ( X  =  U. J  -> 
( Fil `  X
)  =  ( Fil `  U. J ) )
87eleq2d 2537 . . . 4  |-  ( X  =  U. J  -> 
( F  e.  ( Fil `  X )  <-> 
F  e.  ( Fil `  U. J ) ) )
96, 8syl5ibrcom 222 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  ( X  =  U. J  ->  F  e.  ( Fil `  X
) ) )
10 filunibas 20114 . . . . 5  |-  ( F  e.  ( Fil `  U. J )  ->  U. F  =  U. J )
116, 10syl 16 . . . 4  |-  ( A  e.  ( J  fClus  F )  ->  U. F  = 
U. J )
12 filunibas 20114 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
1312eqeq1d 2469 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( U. F  =  U. J  <->  X  =  U. J ) )
1411, 13syl5ibcom 220 . . 3  |-  ( A  e.  ( J  fClus  F )  ->  ( F  e.  ( Fil `  X
)  ->  X  =  U. J ) )
159, 14impbid 191 . 2  |-  ( A  e.  ( J  fClus  F )  ->  ( X  =  U. J  <->  F  e.  ( Fil `  X ) ) )
164, 15bitrd 253 1  |-  ( A  e.  ( J  fClus  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767   U.cuni 4245   ` cfv 5586  (class class class)co 6282   Topctop 19158  TopOnctopon 19159   Filcfil 20078    fClus cfcls 20169
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  ax-un 6574
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 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-fbas 18184  df-topon 19166  df-fil 20079  df-fcls 20174
This theorem is referenced by:  fclsopni  20248  fclselbas  20249  fclsss1  20255  fclsss2  20256  fclscf  20258
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