# equality of functions between sets

Denote $E$ and $F$ to be sets and $f:E\longrightarrow F$ a function and $\Gamma=\{(x,y)\in E\times F| y=f(x)\}$. I read that two functions are equal if and only if the have the same $E$, the same $F$, and the same $\Gamma$. I thought that equality of two functions does not depend on the set $F$, we can just take any set containing the image of $f$ and we still have the same function. Can we say that two functions are equal if and only if they have the same domain and the same graph (without mentionning the set $F$)?