reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem Th20:
  for A,B being set, R being Subset of [:A,B:] holds rng(.:R) c= bool rng R
proof
  let A,B be set, R be Subset of [:A,B:];
  let y be object;
    reconsider yy=y as set by TARSKI:1;
  assume y in rng(.:R);
  then consider x be object such that
A1: x in dom(.:R) and
A2: y = (.:R).x by FUNCT_1:def 3;
  reconsider x as set by TARSKI:1;
  y = R.:x by A1,A2,Th19;
  then yy c= rng R by RELAT_1:111;
  hence thesis;
end;
