 Internal Heyting algebra of truth values

The subobject classifier $\Omega$ of a topos carries internal truth, falsity, meet, join and implication. Its generalized elements over $X$ are subobjects of $X$; implication satisfies $R\le (P\Rightarrow Q)$ exactly when $R\cap P\le Q$. Pullback-compatible operations make this an internal Heyting algebra, not merely a structure on global truth values.