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Theorem fclstop 18004
Description: Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
fclstop  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )

Proof of Theorem fclstop
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 eqid 2412 . . 3  |-  U. J  =  U. J
21isfcls 18002 . 2  |-  ( A  e.  ( J  fClus  F )  <->  ( J  e. 
Top  /\  F  e.  ( Fil `  U. J
)  /\  A. s  e.  F  A  e.  ( ( cls `  J
) `  s )
) )
32simp1bi 972 1  |-  ( A  e.  ( J  fClus  F )  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   A.wral 2674   U.cuni 3983   ` cfv 5421  (class class class)co 6048   Topctop 16921   clsccl 17045   Filcfil 17838    fClus cfcls 17929
This theorem is referenced by:  fclstopon  18005  fclsneii  18010  fclsfnflim  18020  flimfnfcls  18021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-fbas 16662  df-fil 17839  df-fcls 17934
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