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;

theorem Th19:
  dom(f*(id X)) = dom f /\ X
proof
  for x being object holds x in dom(f*(id X)) iff x in dom f /\ X
  proof let x be object;
    x in dom(f*(id X)) iff x in dom f & x in X
    proof
      thus x in dom(f*(id X)) implies x in dom f & x in X
      proof
        assume x in dom(f*(id X));
        then
A1:     x in dom((id X)) & (id X).x in dom f by Th11;
        thus thesis by A1,Th17;
      end;
      assume
A2:   x in dom f;
A3:   dom((id X)) = X;
      assume
A4:   x in X;
      then (id X).x in dom f by A2,Th17;
      hence thesis by A4,A3,Th11;
    end;
    hence thesis by XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:2;
end;
