Lemma I.13.19 on Kunen's set theory

Question

I'm not even sure how to approach this problem. Kunen defined for arbitary sets $B$ and cardinals $\theta$, $$ [B]^\theta = \{ x \subseteq B : |x| = \theta \} $$ $$ [B]^{< \theta} = \{ x \subseteq B : |x| \leq \theta \} $$ And the problem (working in ZFC without foundation) I'm stuck on is the second part of lemma I.13.19: if $B$ is infinite and $\aleph_{0} \leq \theta \leq |B|$, then $$ |[B]^\theta | = |B|^\theta $$ And $$ |[B]^{< \theta} | = |B|^{< \theta} $$ Here $B^A$ is the cardinal exponentiation, and $B^{< A} = | \{ f: C \to B : C$ is an ordinal and $C < A\}|$ I've only managed to show the trivial inequalities like $|[B]^\theta | \leq |B|^\theta$. I'm sure the proof is probably pretty simple, but I'm pretty clueless about what I should do next, any hints would be appreciated!