reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  R"Y = {} iff rng R misses Y
proof
  set x = the Element of R"Y;
  thus R"Y = {} implies rng R misses Y
  proof
    assume
A1: R"Y = {};
    assume not thesis;
    then consider y being object such that
A2: y in rng R and
A3: y in Y by XBOOLE_0:3;
    ex x being object st [x,y] in R by A2,XTUPLE_0:def 13;
    hence contradiction by A1,A2,A3,Th123;
  end;
  assume
A4: rng R /\ Y = {};
  assume not thesis;
  then ex y st y in rng R & [x,y] in R & y in Y by Th123;
  hence contradiction by A4,XBOOLE_0:def 4;
end;
