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Theorem neifval 16852
Description: The neighborhood function on the subsets of a topology's base set. (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 16668 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4210 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5761 . . 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 18 . 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 3852 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2346 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3643 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 rexeq 2750 . . . . 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 2796 . . . 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 4114 . . 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 16851 . . 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 5616 . 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 649 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 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    e. cmpt 4093   ` cfv 5271   Topctop 16647   neicnei 16850
This theorem is referenced by:  neif  16853  neival  16855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-nei 16851
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