Understanding formulas for $E(X∣X < c)$

Question

I have $E(X∣X < c)$ where $X$ is a continuous random variable and $c$ is a given positive real value.

According to this question this is equivalent to:

1. $E[X|X<c]=\int_{-\inf}^{c}1-\frac{F(x)}{F(c)}dx$

On this other question we have both:

2. $\mathrm E(X\mid X < c)=\frac{\mathrm E(X\cdot\mathbf 1_{X < c})}{\mathrm P(X\gt c)}$
3. $E(X|X<c) = \frac{\int_x x f(x) I(x<c) dx}{\int_x f(x) I(x<c) dx}$

Yet, starting from the definition of conditional expectation, I couldn't get to show any of the three equations. I am particularly interested in the first one.

Can you help me understand how to get it? (And or the others)

Thanks.