The union of graph G(f) with this other set ?

[operationsres]

A first year real analysis textbook presents the following two definitions (where the second builds off the first.

(1) Definition (Graph of a map)

A and B are sets and $f : A \rightarrow B$ is some map. Then we define the graph of $f$ by $$G(f) := \{(x,f(x)) \in A \times B : x \in A\}$$.(2) Other definition

A and B are sets and $f : A \rightarrow B$ is some map. Further, define for every $y \in B$ the corresponding intersection $G_{fy}$ by $$G_{fy} := G(f) \cap \{(x,y) : x \in A\}$$.

(This then proceeds into a theorem about bijections).

1. The problem I'm having
I completely understand (1) and all notation employed in both (1) and (2). However, I don't understand what (2) is trying to communicate ... It seems to me that $G_{fy} = G(f)$ based on my interpretation of (2), making $G_{fy}$ superfluous.

 

[operationsres]

Sorry... I finally figured it out after I posted. $f(x) \in image(f)$, $y \in B$ and image(f) is not necessarily equal to B.

Feel free to delete.