theorem
  Y <> {} implies rng pr1(X,Y) = X
proof
  set y = the Element of Y;
  assume
A1: Y <> {};
A2: X c= rng pr1(X,Y)
  proof
    let x be object;
    assume
A3: x in X;
    then [x,y] in [:X,Y:] by A1,ZFMISC_1:87;
    then
A4: [x,y] in dom pr1(X,Y) by Def4;
    pr1(X,Y).(x,y) = x by A1,A3,Def4;
    hence thesis by A4,FUNCT_1:def 3;
  end;
  rng pr1(X,Y) c= X by Th43;
  hence thesis by A2,XBOOLE_0:def 10;
end;
