MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-nei Unicode version

Definition df-nei 16830
Description: Define a function on topologies whose value is a map from a subset to its neighborhoods. (Contributed by NM, 11-Feb-2007.)
Assertion
Ref Expression
df-nei  |-  nei  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { y  e. 
~P U. j  |  E. g  e.  j  (
x  C_  g  /\  g  C_  y ) } ) )
Distinct variable group:    g, j, x, y

Detailed syntax breakdown of Definition df-nei
StepHypRef Expression
1 cnei 16829 . 2  class  nei
2 vj . . 3  set  j
3 ctop 16626 . . 3  class  Top
4 vx . . . 4  set  x
52cv 1623 . . . . . 6  class  j
65cuni 3829 . . . . 5  class  U. j
76cpw 3627 . . . 4  class  ~P U. j
84cv 1623 . . . . . . . 8  class  x
9 vg . . . . . . . . 9  set  g
109cv 1623 . . . . . . . 8  class  g
118, 10wss 3154 . . . . . . 7  wff  x  C_  g
12 vy . . . . . . . . 9  set  y
1312cv 1623 . . . . . . . 8  class  y
1410, 13wss 3154 . . . . . . 7  wff  g  C_  y
1511, 14wa 360 . . . . . 6  wff  ( x 
C_  g  /\  g  C_  y )
1615, 9, 5wrex 2546 . . . . 5  wff  E. g  e.  j  ( x  C_  g  /\  g  C_  y )
1716, 12, 7crab 2549 . . . 4  class  { y  e.  ~P U. j  |  E. g  e.  j  ( x  C_  g  /\  g  C_  y ) }
184, 7, 17cmpt 4079 . . 3  class  ( x  e.  ~P U. j  |->  { y  e.  ~P U. j  |  E. g  e.  j  ( x  C_  g  /\  g  C_  y ) } )
192, 3, 18cmpt 4079 . 2  class  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { y  e.  ~P U. j  |  E. g  e.  j  ( x  C_  g  /\  g  C_  y ) } ) )
201, 19wceq 1624 1  wff  nei  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  { y  e. 
~P U. j  |  E. g  e.  j  (
x  C_  g  /\  g  C_  y ) } ) )
Colors of variables: wff set class
This definition is referenced by:  neifval  16831
  Copyright terms: Public domain W3C validator