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Theorem neifval 13502
Description: Value of the neighborhood function on the subsets of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
neifval.1  |-  X  = 
U. J
Assertion
Ref Expression
neifval  |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } ) )
Distinct variable groups:    v, g, x, J    g, X, v, x

Proof of Theorem neifval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . 4  |-  X  = 
U. J
21topopn 13368 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4179 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5739 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  e.  _V )
52, 3, 43syl 17 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )  e.  _V )
6 unieq 3818 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1eqtr4di 2228 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3580 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 rexeq 2673 . . . . 5  |-  ( j  =  J  ->  ( E. g  e.  j 
( x  C_  g  /\  g  C_  v )  <->  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) ) )
108, 9rabeqbidv 2732 . . . 4  |-  ( j  =  J  ->  { v  e.  ~P U. j  |  E. g  e.  j  ( x  C_  g  /\  g  C_  v ) }  =  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } )
118, 10mpteq12dv 4084 . . 3  |-  ( j  =  J  ->  (
x  e.  ~P U. j  |->  { v  e. 
~P U. j  |  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 ) } ) )
12 df-nei 13501 . . 3  |-  nei  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { v  e. 
~P U. j  |  E. g  e.  j  (
x  C_  g  /\  g  C_  v ) } ) )
1311, 12fvmptg 5590 . 2  |-  ( ( J  e.  Top  /\  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } )  e.  _V )  ->  ( nei `  J
)  =  ( x  e.  ~P X  |->  { v  e.  ~P X  |  E. g  e.  J  ( x  C_  g  /\  g  C_  v ) } ) )
145, 13mpdan 421 1  |-  ( J  e.  Top  ->  ( nei `  J )  =  ( x  e.  ~P X  |->  { v  e. 
~P X  |  E. g  e.  J  (
x  C_  g  /\  g  C_  v ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456   {crab 2459   _Vcvv 2737    C_ wss 3129   ~Pcpw 3575   U.cuni 3809    |-> cmpt 4063   ` cfv 5214   Topctop 13357   neicnei 13500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-top 13358  df-nei 13501
This theorem is referenced by:  neif  13503  neival  13505
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