reserve X,Y for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,h for Function;

theorem
  x in dom f & x in X iff [x,f.x] in f|X
proof
  x in dom f iff [x,f.x] in f by FUNCT_1:def 2,XTUPLE_0:def 12;
  hence thesis by RELAT_1:def 11;
end;
