# Equation for likelihood in logistic regression

I read the likelihood is defined in logistic regression as the probability $$L(w) = P(y|x, w) = \prod P(y^i| x^i,w) = \prod (\sigma(z^i))^{y^i}(1-\sigma(z^i))^{(1-y^i)}$$ and the log of the last equation is:

$$log(L(w)) = \sum y^i log((\sigma(z^i)) + (1-y^i)log(1-\sigma(z^i))$$

I understand (independent probabilities) $$L(w) = P(y|x, w) = \prod P(y^i| x^i,w)$$ and I understand that from $$L(w) = \prod (\sigma(z^i))^{y^i}(1-\sigma(z^i))^{(1-y^i)}$$ the log is (basic log properties) $$log(L(w)) = \sum y^i log((\sigma(z^i)) + (1-y^i)log(1-\sigma(z^i))$$ However, how do I get $$\prod P(y^i| x^i,w) = \prod (\sigma(z^i))^{y^i}(1-\sigma(z^i))^{(1-y^i)}$$ This basically means that $$P(y| x,w) = (\sigma(z))^{y}(1-\sigma(z))^{(1-y)}$$ and I just don't see that.

By the definition of the logistic regression model $\mathrm P(y = 1 | x,w) = \sigma\left(\left<w, x\right>\right)$ thus $\mathrm P(y = 0 | x,w) = 1- \sigma\left(\left<w, x\right>\right)$
$\mathrm P(y | x,w) = \sigma\left(\left<w, x\right>\right)^y \left[ 1- \sigma \left( \left<w, x \right> \right) \right]^{1-y}$