The domain of discourse is reality.
Explanation. Def. 2 is used in the cultural ideas of political relevance, while not entirely correct or comprehensive.[ii] Otherwise, people would not call themselves women and thus have no reason to argue about within politics.
Proposition 1. EVERY TRANSGENDER WOMAN IS A PERSON WHOSE GENDER IDENTITY CULTURALLY CONFORM TO THE FEMALE SEX.
Proof. If every person whose gender identity does not culturally conform to their associated sex assigned at birth is transgender, then every transgender woman is a person whose gender identity culturally conform to the female sex [Axiom 1]. Every person whose gender identity does not culturally conform to their associated sex assigned at birth is transgender [Def. 2].
Proposition 2. EVERY TRANSGENDER WOMAN IS A WOMAN.
Proof. Virtually every person whose gender identity culturally conform to the female sex is a woman [Axiom 2]. Every transgender woman is a person whose gender identity culturally conform to the female sex [1].
Remark. Someone might question why the proof is reasoned inductively rather than deductively: Any construction of a deductive argument could not be satisfying. The main reason is that the argument around this topic that gender is synonymous with sex is, correctly, incoherent. Furthermore, another level of why transgender women are women is changing the definition of woman based on utility, which is correct. However, trying to extrapolate a definition from the APA[iv] does not seem enough to justify this claim-at least, from my full effort.
"Gender identity is a component of gender that describes a person's psychological sense of their gender."
"Transgender is used as an adjective to refer to persons whose gender identity, expression, and/or role does not conform to what is culturally associated with their sex assigned at birth."
"Among U.S. adults, 0.5% (about 1.3 million adults) identify as transgender," meaning 99.5% of people are cisgender-what the opposition would call women or men.
Women are people with the attitudes, feelings, and behaviors that a given culture associates with a person's biological female sex. Extrapolated from: "Gender refers to the attitudes, feelings, and behaviors that a given culture associates with a person's biological sex."