Monotonicity of second price auctions with bidder-specific reserve prices I am having difficulty understanding the problem below:

Given CDFs $F_1, ..., F_n$, prove that the allocation rule of a second-price auction with bidder-specific reserve prices is monotone. The item is rewarded to the bidder with the highest bid that meets his reserve price. The reserve price of bidder $i$ is $r_i = _i^{-1}(t)$ where $_i$ is the virtual valuation function

$$\phi_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$$

and $t$ is chosen such that 

$$Pr\left[\max_i _i(v_i) \ge t\right] = 1/2.$$ 
Since the virtual valuations are determined before the auction (if I understand it correctly), how does the reserve prices affect the monotonicity of the allocation rule? The behavior is still the same: if you keep on increasing your bid, you'll have a higher chance of getting the price. How should I go about this?
Some thoughts. If this was a SPA without a reserve price, the monotonicity of the allocation rule is trivial. When the reserve price is uniform across bidders then the allocation rule is clearly weakly monotonic now; since for low bids, increasing them might not change the probability of allocation. The only thing that makes this problem different is that the reserve prices are bidder-specific. So one possibility is that you only need to show that the logic for the previous monotonicity is still valid in this setup. Probably while trying to show that the logic still goes through you will find out if there is some extra argument you will have to make in order to still have the monotonicity. 

For example, I might worry that by increasing my bid I might increase my own reserve price, or decrease the reserve price of some other bidder and thus reduce my probability of being allocated the good.