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Theorem flimtopon 20199
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
flimtopon  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )

Proof of Theorem flimtopon
StepHypRef Expression
1 flimtop 20194 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  J  e.  Top )
2 istopon 19186 . . . 4  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
32baib 898 . . 3  |-  ( J  e.  Top  ->  ( J  e.  (TopOn `  X
)  <->  X  =  U. J ) )
41, 3syl 16 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  (TopOn `  X )  <->  X  =  U. J ) )
5 eqid 2460 . . . . 5  |-  U. J  =  U. J
65flimfil 20198 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
7 fveq2 5857 . . . . 5  |-  ( X  =  U. J  -> 
( Fil `  X
)  =  ( Fil `  U. J ) )
87eleq2d 2530 . . . 4  |-  ( X  =  U. J  -> 
( F  e.  ( Fil `  X )  <-> 
F  e.  ( Fil `  U. J ) ) )
96, 8syl5ibrcom 222 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( X  =  U. J  ->  F  e.  ( Fil `  X
) ) )
10 filunibas 20110 . . . . 5  |-  ( F  e.  ( Fil `  U. J )  ->  U. F  =  U. J )
116, 10syl 16 . . . 4  |-  ( A  e.  ( J  fLim  F )  ->  U. F  = 
U. J )
12 filunibas 20110 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  U. F  =  X )
1312eqeq1d 2462 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( U. F  =  U. J  <->  X  =  U. J ) )
1411, 13syl5ibcom 220 . . 3  |-  ( A  e.  ( J  fLim  F )  ->  ( F  e.  ( Fil `  X
)  ->  X  =  U. J ) )
159, 14impbid 191 . 2  |-  ( A  e.  ( J  fLim  F )  ->  ( X  =  U. J  <->  F  e.  ( Fil `  X ) ) )
164, 15bitrd 253 1  |-  ( A  e.  ( J  fLim  F )  ->  ( J  e.  (TopOn `  X )  <->  F  e.  ( Fil `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   U.cuni 4238   ` cfv 5579  (class class class)co 6275   Topctop 19154  TopOnctopon 19155   Filcfil 20074    fLim cflim 20163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-fbas 18180  df-top 19159  df-topon 19162  df-nei 19358  df-fil 20075  df-flim 20168
This theorem is referenced by:  fclsfnflim  20256
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