reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function
  of Y,X,
  x,x1,x2 for Element of X;
reserve y for Element of Y;
reserve Y for non empty set,
  F for BinOp of X,
  f for Function of Y,X,
  x for Element of X,
  y for Element of Y;
reserve a,b,c for set;
reserve x,y,z for object;
reserve Y for set,
        f,g for Function of Y,X,
        x for Element of X,
        y for Element of Y;

theorem
 for x,y,A being set st x in A
 holds (x .--> y)|A = x .--> y
proof let x,y,A be set;
  assume x in A;
  then dom(x .--> y) c= A by ZFMISC_1:31;
  hence thesis by RELAT_1:68;
end;
