I just edited my whole question since i think it was a bit messy.
Here is my question.
Let $K$ be a separable compact metric space and $S\subset C(K,\mathbb{C})$.
Let $S$ be closed,bounded,uniformly equicontinuous on $K$, sequentially compact, totally bounded and complete.
Then is $S$ compact? (in ZF)
Thank you in advance!