
theorem Th1:
  for f being Function, Y being set holds dom(Y|`f) = f"Y
proof
  let f be Function;
  let Y be set;
  for x being object holds x in dom(Y|`f) iff x in f"Y
  proof
    let x be object;
    hereby
      assume x in dom(Y|`f);
      then consider y be object such that
A1:   [x,y] in Y|`f by XTUPLE_0:def 12;
      y in Y & [x,y] in f by A1,RELAT_1:def 12;
      hence x in f"Y by RELAT_1:def 14;
    end;
    assume x in f"Y;
    then consider y be object such that
A2: [x,y] in f & y in Y by RELAT_1:def 14;
    [x,y] in Y|`f by A2,RELAT_1:def 12;
    hence x in dom(Y|`f) by XTUPLE_0:def 12;
  end;
  hence dom(Y|`f) = f"Y by TARSKI:2;
end;
