reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;
reserve e,u for object,
  A for Subset of X;

theorem
  X in dom f implies rng(f|{X}) ={f.X}
proof
A1: X in {X} by TARSKI:def 1;
  assume X in dom f;
  then
A2: X in dom(f|{X}) by A1,RELAT_1:57;
  thus rng(f|{X}) c= {f.X} by Th96;
  let x be object;
  assume x in {f.X};
  then x = f.X by TARSKI:def 1;
  then x = (f|{X}).X by A2,Th46;
  hence thesis by A2,Def3;
end;
