I'm reading the paper "The relevance of convex analysis in the study of monotonicity" by Jean-Paul Penot. (A really nice paper!) There is one "classical result of convex analysis" being used:
For Banach spaces $X, Y$, a linear and continuous map $A:X\rightarrow Y$ and $\varphi:Y\rightarrow\mathbb{R}\cup \{ + \infty \} $ convex, proper and lower-semicontinuous it holds
$$ (\varphi\circ A)^* = \min \{ \varphi ^* (y^*) : A^* (y^*) = x^* \} $$
if $\mathbb{R}_+(\rm{dom}\ \varphi - \rm{Im}(A))=Y$ (where "min" means that the infimum is attained when it is finite).
Can someone give a proof or provide a good reference? (I am sure that there are generalizations but this level of generality is totally sufficient for me.)