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Theorem trnei 17603
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 17598 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 16680 . . . 4  |-  ( J  e.  (TopOn `  Y
)  ->  J  e.  Top )
213ad2ant1 976 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  Top )
3 simp2 956 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_  Y )
4 toponuni 16681 . . . . 5  |-  ( J  e.  (TopOn `  Y
)  ->  Y  =  U. J )
543ad2ant1 976 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  Y  =  U. J )
63, 5sseqtrd 3227 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_ 
U. J )
7 simp3 957 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  Y )
87, 5eleqtrd 2372 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  U. J )
9 eqid 2296 . . . 4  |-  U. J  =  U. J
109neindisj2 16876 . . 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 1182 . 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 955 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  (TopOn `  Y )
)
137snssd 3776 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  C_  Y )
14 snnzg 3756 . . . . 5  |-  ( P  e.  Y  ->  { P }  =/=  (/) )
15143ad2ant3 978 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  =/=  (/) )
16 neifil 17591 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  { P }  C_  Y  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
1712, 13, 15, 16syl3anc 1182 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  (
( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
18 trfil2 17598 . . 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 642 . 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 247 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 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   clsccl 16771   neicnei 16850   Filcfil 17556
This theorem is referenced by:  limcflflem  19246  islimrs3  25684  islimrs4  25685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-rest 13343  df-top 16652  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-fbas 17536  df-fil 17557
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