
theorem Th5:
  for f being empty Function holds .:f = {} .--> {}
proof
  let f be empty Function;
  A1: dom(.:f) = bool dom f by FUNCT_3:def 1
    .= {{}} by ZFMISC_1:1;
  then A2: dom(.:f) = dom {[{},{}]} by RELAT_1:9
    .= dom({} .--> {}) by FUNCT_4:82;
  now
    let x be object;
    assume x in dom(.:f);
    then A3: x = {} by A1, TARSKI:def 1;
    hence (.:f).x = {} by FUNCT_3:8
      .= ({} .--> {}).x by A3, FUNCOP_1:72;
  end;
  hence thesis by A2, FUNCT_1:2;
end;
