Can I set pin 13 to output 5v, then put arduino mini pro to sleep?
Given the absence of clarity on the driving state of pins during sleep, consider connecting Pin 13 to an SR Latch or the gate of a MOSFET.

An SR Latch will hold it's output high or low until a high is presented on the Set or Reset inputs, so connect Pin 13 to the Set input, with a pull-down resistor -- even if the MCU in sleep mode allows the voltage output to drain away to 0v, the SR Latch will maintain it's output.

A MOSFET gate has a non-trivial capacitance which can hold a charge if the gate's driver goes high-impedance, thus holding the gate open after the driver is disconnected.