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Theorem trnei 17847
Description: The trace, over a set  A, of the filter of the neighborhoods of a point  P is a filter iff  P belongs to the closure of  A. (This is trfil2 17842 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
trnei  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  ( P  e.  ( ( cls `  J ) `  A )  <->  ( (
( nei `  J
) `  { P } )t  A )  e.  ( Fil `  A ) ) )

Proof of Theorem trnei
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 topontop 16916 . . . 4  |-  ( J  e.  (TopOn `  Y
)  ->  J  e.  Top )
213ad2ant1 978 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  Top )
3 simp2 958 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_  Y )
4 toponuni 16917 . . . . 5  |-  ( J  e.  (TopOn `  Y
)  ->  Y  =  U. J )
543ad2ant1 978 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  Y  =  U. J )
63, 5sseqtrd 3329 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_ 
U. J )
7 simp3 959 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  Y )
87, 5eleqtrd 2465 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  U. J )
9 eqid 2389 . . . 4  |-  U. J  =  U. J
109neindisj2 17112 . . 3  |-  ( ( J  e.  Top  /\  A  C_  U. J  /\  P  e.  U. J )  ->  ( P  e.  ( ( cls `  J
) `  A )  <->  A. v  e.  ( ( nei `  J ) `
 { P }
) ( v  i^i 
A )  =/=  (/) ) )
112, 6, 8, 10syl3anc 1184 . 2  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  ( P  e.  ( ( cls `  J ) `  A )  <->  A. v  e.  ( ( nei `  J
) `  { P } ) ( v  i^i  A )  =/=  (/) ) )
12 simp1 957 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  (TopOn `  Y )
)
137snssd 3888 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  C_  Y )
14 snnzg 3866 . . . . 5  |-  ( P  e.  Y  ->  { P }  =/=  (/) )
15143ad2ant3 980 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  =/=  (/) )
16 neifil 17835 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  { P }  C_  Y  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
1712, 13, 15, 16syl3anc 1184 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  (
( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
18 trfil2 17842 . . 3  |-  ( ( ( ( nei `  J
) `  { P } )  e.  ( Fil `  Y )  /\  A  C_  Y
)  ->  ( (
( ( nei `  J
) `  { P } )t  A )  e.  ( Fil `  A )  <->  A. v  e.  (
( nei `  J
) `  { P } ) ( v  i^i  A )  =/=  (/) ) )
1917, 3, 18syl2anc 643 . 2  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  (
( ( ( nei `  J ) `  { P } )t  A )  e.  ( Fil `  A )  <->  A. v  e.  (
( nei `  J
) `  { P } ) ( v  i^i  A )  =/=  (/) ) )
2011, 19bitr4d 248 1  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  ( P  e.  ( ( cls `  J ) `  A )  <->  ( (
( nei `  J
) `  { P } )t  A )  e.  ( Fil `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651    i^i cin 3264    C_ wss 3265   (/)c0 3573   {csn 3759   U.cuni 3959   ` cfv 5396  (class class class)co 6022   ↾t crest 13577   Topctop 16883  TopOnctopon 16884   clsccl 17007   neicnei 17086   Filcfil 17800
This theorem is referenced by:  cnextfun  18018  cnextfvval  18019  cnextf  18020  cnextcn  18021  cnextfres  18022  cnextucn  18256  ucnextcn  18257  limcflflem  19636  rrhre  24185
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-rest 13579  df-fbas 16625  df-top 16888  df-topon 16891  df-cld 17008  df-ntr 17009  df-cls 17010  df-nei 17087  df-fil 17801
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