In the mishna there is a machlokes whether one is believed to claim that his son is a mamzer. The Shulchan Aruch (Even Haezer 4:29) says that although we pasken like R' Yehuda that a father is believed that his son is a mamzer, a mother is not believed. R' Moshe has many teshuvos where the mother claimed her son is a mamzer because she was married prior to this marriage, and never received a gett. R' Moshe (E.H. 4:23:2) ignored all these claims, and held that the mother is not even believed to create a safeik, because the din of "yakir" is only on the father. In regard to believing the father, R' Moshe writes (4:23:1) that it only applies when we know that he is father such as a case where he is married to the mother. But if the fact that he is the father is purely based on his claim, then he is not believed. He proves this from the Rambam (Issurei Biah 15:16):
אבל האב שהוחזק שזה בנו ואמר בני זה ממזר הוא נאמן
Clearly, the Rambam implies that only if it is "huchzak" that this is his son, is he believed about him to make him a mamzer. But, if the we have not chazaka that this is the father because he is not married to the mother and the son didn't live with him, he is not believed to create the status of even a safeik mamzer. R' Moshe then launches into a big discussion based on the Rambam that a man is believed to claim about someone in the street "this is my son and he is a bechor". Since both the right to claim he is a bechor and the right to claim he is a mamzer come from the same source - "yakir", why by the bechor is he believed even on someone who is not muchzak as his son?