MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trnei Structured version   Unicode version

Theorem trnei 19607
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 19602 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 18673 . . . 4  |-  ( J  e.  (TopOn `  Y
)  ->  J  e.  Top )
213ad2ant1 1009 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  Top )
3 simp2 989 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_  Y )
4 toponuni 18674 . . . . 5  |-  ( J  e.  (TopOn `  Y
)  ->  Y  =  U. J )
543ad2ant1 1009 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  Y  =  U. J )
63, 5sseqtrd 3503 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  A  C_ 
U. J )
7 simp3 990 . . . 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 18869 . . 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 1219 . 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 988 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  J  e.  (TopOn `  Y )
)
137snssd 4129 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  C_  Y )
14 snnzg 4103 . . . . 5  |-  ( P  e.  Y  ->  { P }  =/=  (/) )
15143ad2ant3 1011 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  { P }  =/=  (/) )
16 neifil 19595 . . . 4  |-  ( ( J  e.  (TopOn `  Y )  /\  { P }  C_  Y  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
1712, 13, 15, 16syl3anc 1219 . . 3  |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  (
( nei `  J
) `  { P } )  e.  ( Fil `  Y ) )
18 trfil2 19602 . . 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 661 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    i^i cin 3438    C_ wss 3439   (/)c0 3748   {csn 3988   U.cuni 4202   ` cfv 5529  (class class class)co 6203   ↾t crest 14482   Topctop 18640  TopOnctopon 18641   clsccl 18764   neicnei 18843   Filcfil 19560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-rest 14484  df-fbas 17949  df-top 18645  df-topon 18648  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-fil 19561
This theorem is referenced by:  cnextfun  19778  cnextfvval  19779  cnextf  19780  cnextcn  19781  cnextfres  19782  cnextucn  20020  ucnextcn  20021  limcflflem  21498  rrhre  26615
  Copyright terms: Public domain W3C validator