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Theorem neif 18703
Description: The neighborhood function is a function of the subsets of a topology's base set. (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 18518 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4475 . . . . 5  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4441 . . . . 5  |-  ( ~P X  e.  _V  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
52, 3, 43syl 20 . . . 4  |-  ( J  e.  Top  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
65ralrimivw 2799 . . 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 2442 . . . 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 5536 . . 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 16 . 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 18702 . . 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 5500 . 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 232 1  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 2971    C_ wss 3327   ~Pcpw 3859   U.cuni 4090    e. cmpt 4349    Fn wfn 5412   ` cfv 5417   Topctop 18497   neicnei 18700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-top 18502  df-nei 18701
This theorem is referenced by:  neiss2  18704
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