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Theorem neif 19727
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 19541 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4640 . . . . 5  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4606 . . . . 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 2872 . . 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 2457 . . . 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 5713 . . 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 19726 . . 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 5677 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   U.cuni 4251    |-> cmpt 4515    Fn wfn 5589   ` cfv 5594   Topctop 19520   neicnei 19724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19525  df-nei 19725
This theorem is referenced by:  neiss2  19728
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