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Theorem trnei 20559
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 20554 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 19594 . . . 4  |-  ( J  e.  (TopOn `  Y
)  ->  J  e.  Top )
213ad2ant1 1015 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  Top )
3 simp2 995 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_  Y )
4 toponuni 19595 . . . . 5  |-  ( J  e.  (TopOn `  Y
)  ->  Y  =  U. J )
543ad2ant1 1015 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  Y  =  U. J )
63, 5sseqtrd 3525 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_ 
U. J )
7 simp3 996 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  Y )
87, 5eleqtrd 2544 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  U. J )
9 eqid 2454 . . . 4  |-  U. J  =  U. J
109neindisj2 19791 . . 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 1226 . 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 994 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  (TopOn `  Y )
)
137snssd 4161 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  C_  Y )
14 snnzg 4133 . . . . 5  |-  ( P  e.  Y  ->  { P }  =/=  (/) )
15143ad2ant3 1017 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  =/=  (/) )
16 neifil 20547 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  { P }  C_  Y  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
1712, 13, 15, 16syl3anc 1226 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  (
( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
18 trfil2 20554 . . 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 659 . 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 256 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   U.cuni 4235   ` cfv 5570  (class class class)co 6270   ↾t crest 14910   Topctop 19561  TopOnctopon 19562   clsccl 19686   neicnei 19765   Filcfil 20512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-rest 14912  df-fbas 18611  df-top 19566  df-topon 19569  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-fil 20513
This theorem is referenced by:  cnextfun  20730  cnextfvval  20731  cnextf  20732  cnextcn  20733  cnextfres  20734  cnextucn  20972  ucnextcn  20973  limcflflem  22450  rrhre  28233
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