Late May Yosemite Visit, help

Hello, I am new to the forum.
I have recently obtained a wilderness permit for Yosemite for 4 days during the end of May (which i am now starting to doubt). I guess I should have done a little more research before I booked them. A friend and I, who both possess a good amount of hiking experience and fitness level, are planning on starting our hiking from Tenaya Lake. Before I booked the permit I looked up the history of when Tioga Road has opened seems like it may or may not be open this time of year. Since I am a little late to the reservation game that opened up 2 months ago, I decided I might just gamble because permits are probably all taken up. So i obtained a permit and was excited, but then I started doing a bit more research.
I was hoping some people could shed some light on this for me. Will this be doable or should I try to rebook our trip/go somewhere else? I hear that there is a stream crossing at Tenaya Lake that will be high and that there is a possibility of snow. I have all the necessary outdoor equipment to deal with winter hiking, I was just wondering if others think it is an ok idea, as I do not like to leave a lot to chance when it comes to hiking in the winter time.
Any information on the topic will be welcome. While I have looked at various websites on weather in May, I am from the East Coast so to me May is a pretty good time to hike with minimal snow (if any), so I am probably naive on what May is like on the West Coast or in Yosemite. To me California winters are seen as mild compared to what I usually deal with. My friend is from California, but from the San Luis Obispo area and has never been to Yosemite. Hopefully that background helps out a little in your responses. Thanks!
I have recently obtained a wilderness permit for Yosemite for 4 days during the end of May (which i am now starting to doubt). I guess I should have done a little more research before I booked them. A friend and I, who both possess a good amount of hiking experience and fitness level, are planning on starting our hiking from Tenaya Lake. Before I booked the permit I looked up the history of when Tioga Road has opened seems like it may or may not be open this time of year. Since I am a little late to the reservation game that opened up 2 months ago, I decided I might just gamble because permits are probably all taken up. So i obtained a permit and was excited, but then I started doing a bit more research.
I was hoping some people could shed some light on this for me. Will this be doable or should I try to rebook our trip/go somewhere else? I hear that there is a stream crossing at Tenaya Lake that will be high and that there is a possibility of snow. I have all the necessary outdoor equipment to deal with winter hiking, I was just wondering if others think it is an ok idea, as I do not like to leave a lot to chance when it comes to hiking in the winter time.
Any information on the topic will be welcome. While I have looked at various websites on weather in May, I am from the East Coast so to me May is a pretty good time to hike with minimal snow (if any), so I am probably naive on what May is like on the West Coast or in Yosemite. To me California winters are seen as mild compared to what I usually deal with. My friend is from California, but from the San Luis Obispo area and has never been to Yosemite. Hopefully that background helps out a little in your responses. Thanks!