theorem
  X <> {} implies rng pr2(X,Y) = Y
proof
  set x = the Element of X;
  assume
A1: X <> {};
A2: Y c= rng pr2(X,Y)
  proof
    let y be object;
    assume
A3: y in Y;
    then [x,y] in [:X,Y:] by A1,ZFMISC_1:87;
    then
A4: [x,y] in dom pr2(X,Y) by Def5;
    pr2(X,Y).(x,y) = y by A1,A3,Def5;
    hence thesis by A4,FUNCT_1:def 3;
  end;
  rng pr2(X,Y) c= Y by Th45;
  hence thesis by A2,XBOOLE_0:def 10;
end;
