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Theorem neif 17156
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 16971 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4375 . . . . 5  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4345 . . . . 5  |-  ( ~P X  e.  _V  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
52, 3, 43syl 19 . . . 4  |-  ( J  e.  Top  ->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) }  e.  _V )
65ralrimivw 2782 . . 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 2435 . . . 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 5563 . . 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 17155 . . 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 5528 . 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 224 1  |-  ( J  e.  Top  ->  ( nei `  J )  Fn 
~P X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   U.cuni 4007    e. cmpt 4258    Fn wfn 5441   ` cfv 5446   Topctop 16950   neicnei 17153
This theorem is referenced by:  neiss2  17157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-top 16955  df-nei 17154
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