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Theorem neifval 12204
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 12070 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4072 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5611 . . 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 3713 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2166 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3483 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 rexeq 2602 . . . . 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 2653 . . . 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 3978 . . 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 12203 . . 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 5463 . 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 415 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 103    = wceq 1314    e. wcel 1463   E.wrex 2392   {crab 2395   _Vcvv 2658    C_ wss 3039   ~Pcpw 3478   U.cuni 3704    |-> cmpt 3957   ` cfv 5091   Topctop 12059   neicnei 12202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-top 12060  df-nei 12203
This theorem is referenced by:  neif  12205  neival  12207
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