ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  neif Unicode version

Theorem neif 12347
Description: The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neif  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )

Proof of Theorem neif
Dummy variables  g  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . . 6  |-  X  = 
U. J
21topopn 12212 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4111 . . . . 5  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4078 . . . . 5  |-  ( ~P X  e.  _V  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
52, 3, 43syl 17 . . . 4  |-  ( J  e.  Top  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
65ralrimivw 2509 . . 3  |-  ( J  e.  Top  ->  A. x  e.  ~P  X { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
7 eqid 2140 . . . 4  |-  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )
87fnmpt 5256 . . 3  |-  ( A. x  e.  ~P  X { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V  ->  ( x  e. 
~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  Fn  ~P X
)
96, 8syl 14 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  Fn  ~P X )
101neifval 12346 . . 3  |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } ) )
1110fneq1d 5220 . 2  |-  ( J  e.  Top  ->  (
( nei `  J
)  Fn  ~P X  <->  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  Fn  ~P X ) )
129, 11mpbird 166 1  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   _Vcvv 2689    C_ wss 3075   ~Pcpw 3514   U.cuni 3743    |-> cmpt 3996    Fn wfn 5125   ` cfv 5130   Topctop 12201   neicnei 12344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-top 12202  df-nei 12345
This theorem is referenced by:  neiss2  12348
  Copyright terms: Public domain W3C validator