Two definitions of conditional cumulative distribution function. Why are they equivalent?

Question

I quote definition of conditional cumulative distribution function (for two random variables $X$ and $Y$) from two different sources ($F$ denotes c.d.f. while $f$ denotes p.d.f.).

1. $P\left[X\leq x| Y\in\left(y, y+dy\right)\right]=\dfrac{\left[F\left(x,y+dy\right)-F(x,y)\right]/dy}{\left[F_Y\left(y+dy\right)-F_Y(y)\right]/dy}\hspace{0.5cm}\stackrel{dy\rightarrow0}\rightarrow\hspace{0.5cm}\dfrac{\partial{F(x,y)/\partial{y}}}{f_Y(y)}$;
2. $P\left[X\leq x| Y\in\left(y, y+dy\right)\right]=\dfrac{\displaystyle\int_{u=-\infty}^{x}f(u,y)dudy}{f_Y(y)dy}=\displaystyle\int_{u=-\infty}^{x}\dfrac{f(u,y)du}{f_Y(y)}$
   
   
   

My question is: why $\dfrac{\partial{F(x,y)/\partial{y}}}{f_Y(y)}=\displaystyle\int_{u=-\infty}^{x}\dfrac{f(u,y)du}{f_Y(y)}$?