reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;

theorem
  for A being (Element of Fin X), B being set for f being Function of X,
  Y st for x being Element of X holds x in A implies f.x in B holds f.:A c= B
proof
  let A be (Element of Fin X), B be set;
  let f be Function of X,Y such that
A1: for x being Element of X holds x in A implies f.x in B;
  let x be object;
  assume x in f.:A;
  then consider y being object such that
  y in dom f and
A2: y in A and
A3: x = f.y by FUNCT_1:def 6;
  reconsider y as Element of X by A2,Th6;
  x = f.y by A3;
  hence thesis by A1,A2;
end;
