有点难度的英语语法请教,First thing I'm going to do is lock them up.

有点难度的英语语法请教,First thing I'm going to do is lock them up.
请看以下这句英语口语：
First thing I'm going to do is lock them up.
[翻译：我要做的第一件事是把他们锁起来.]
以下我分析了该句子的语法成份：
I'm going to do 是定语从句修饰First thing的,这里省略了which/that.
而lock them up是表语从句.
但我觉得这个句子有错,因为lock them up是表语从句,应该在其前面加that才对.
比如说：My worry is that he doesn't work.
你看,he doesn't work是表语从句,前面有个that.
所以我觉得要加that,正确的写法应该是：
First thing I'm going to do is that lock them up.