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Theorem trnei 19307
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 19302 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 18373 . . . 4  |-  ( J  e.  (TopOn `  Y
)  ->  J  e.  Top )
213ad2ant1 1002 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  Top )
3 simp2 982 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_  Y )
4 toponuni 18374 . . . . 5  |-  ( J  e.  (TopOn `  Y
)  ->  Y  =  U. J )
543ad2ant1 1002 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  Y  =  U. J )
63, 5sseqtrd 3380 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_ 
U. J )
7 simp3 983 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  Y )
87, 5eleqtrd 2509 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  P  e.  U. J )
9 eqid 2433 . . . 4  |-  U. J  =  U. J
109neindisj2 18569 . . 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 1211 . 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 981 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  (TopOn `  Y )
)
137snssd 4006 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  C_  Y )
14 snnzg 3980 . . . . 5  |-  ( P  e.  Y  ->  { P }  =/=  (/) )
15143ad2ant3 1004 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  =/=  (/) )
16 neifil 19295 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  { P }  C_  Y  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
1712, 13, 15, 16syl3anc 1211 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  (
( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
18 trfil2 19302 . . 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 654 . 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705    i^i cin 3315    C_ wss 3316   (/)c0 3625   {csn 3865   U.cuni 4079   ` cfv 5406  (class class class)co 6080   ↾t crest 14342   Topctop 18340  TopOnctopon 18341   clsccl 18464   neicnei 18543   Filcfil 19260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-rest 14344  df-fbas 17658  df-top 18345  df-topon 18348  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-fil 19261
This theorem is referenced by:  cnextfun  19478  cnextfvval  19479  cnextf  19480  cnextcn  19481  cnextfres  19482  cnextucn  19720  ucnextcn  19721  limcflflem  21197  rrhre  26301
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