Is my proof of "If $G_1$ and $G_2$ are isomorphic then $|V_1|=|V_2|$ and $|E_1|=|E_2|$ " correct?

Question

I'd like to prove

"If $G_1(V_1,E_1)$ and $G_2(V_2,E_2)$ are isomorphic graphs then $|V_1|=|V_2|$ and $|E_1|=|E_2|$"

$V$=vertex set

$E$=edge set

The notes I'm working through mention that this proof is straightforward but I'd like to make sure that my proof is rigorous enough.

Proof: $G_1(V_1,E_1)$ and $G_2(V_2,E_2)$ be isomorphic graphs.

$\implies$ there exists a bijection $\alpha: V_1 \to V_2$ such that the edge $(\alpha(a),\alpha(b)) \in E_2$ if and only if $(a,b) \in E_1$ [definition of two graphs being isomorphic]

to prove:$|V_1|=|V_2|$:

As there exists a bijection between V_1 and V_2, we can conclude that their cardinalities are the same i.e.$|V_1|=|V_2|$

to prove:$|E_1|=|E_2|$: now as by the definition above, $\alpha(a),\alpha(b)) \in E_2$ if and only if $(a,b) \in E_1$, where $\alpha$ is a bijection.

Thus we have a bijection between the edge sets which implies that $|E_1|=|E_2|$.

Answer 1

That's right! Since $G_1$ and $G_2$ are isomorphic, there's a bijection between their underlying sets (namely, the sets of vertices) that preserves the structure of the graph (namely, the edge connections). That bijection immediately gives you the cardinality claim for the sets of vertices. And that bijection will generate a bijection between the set of edges, which gives you the cardinality claim for the sets of edges.