JS check if checkbox is checked keep initial value

i have a form to update personnal data with multiple checkboxes who can be initially checked or not.

<form method="post" id="up_mission">
    <input checked type="checkbox"  id="first">
    <input type="checkbox"  id="second">
    <button class="btn btn-default" type="submit" id="update_mission<?php echo $id_mission ?>"</button>
</form>

and the JS:

$(function() {
        $("form[id^='up_mission']").submit(function() {
         var value1 = document.getElementById("first").checked;
         var value2 = document.getElementById("second").checked;
         $.post("update_mission.php", {value1: value1, value2:value2}, functon(data) { console.log(data) });
    });
});

in update_mission.php there is a simple sql querry to update with news values and i also do var_dump($_POST); and the values printed for first and second are always the ones that i initially put in the it doesn't matters if i uncheck or check the checkbox.

How can i fix it please?