reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem Th23:
  .:R.:{_{X}_} = {} iff X = {}
proof
  thus .:R.:{_{X}_} = {} implies X = {}
  proof
    assume .:R.:{_{X}_} = {};
    then dom .:R misses {_{X}_} by RELAT_1:118;
    then
A1: bool A misses {_{X}_} by Def1;
    {_{X}_} c= bool A
    proof
      let y be object;
       reconsider yy=y as set by TARSKI:1;
      assume y in {_{X}_};
      then consider x being object such that
A2:   y = {x} and
A3:   x in X by Th1;
A4:   x in A by A3;
      yy c= A
      by A2,A4,TARSKI:def 1;
      hence thesis;
    end;
    then
A5: bool A /\ {_{X}_} = {_{X}_} by XBOOLE_1:28;
    assume X <> {};
    then {_{X}_} <> {} by Th2;
    hence thesis by A1,A5;
  end;
  assume X = {};
  then {_{X}_} = {} by Th2;
  hence thesis;
end;
