gt.geometric topology - Spanning discs in contractible 2-d complexes

I'm pretty sure this has an easy solution, but I can't seem to find it.

Let $X$ be a contractible $2$-dimensional CW-complex, let $gamma$ be an embedded loop in $X$, and let $f : D^2 rightarrow X$ be an embedding of a disc in $X$ which maps the boundary of $D$ to $gamma$.

My question is the following. Let $f' : D^2 rightarrow X$ be a continuous map of a disc into $X$ which takes the boundary of $D$ to $gamma$. Must we then have $f(D^2) subset f'(D^2)$ ? I'm pretty sure that the answer is yes, but I can't seem to prove it.

Of course, this has an obvious generalization to higher dimensional complexes, and I'd be interested in that too.