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I am trying to prove exercise 4.27 in "Sets for Mathematics" by Lawvere and Rosebrugh. The problem is to prove the following equivalences:

For a relation $(p_0,p_1):R \rightarrow X \times X$ on $X$ the following are equivalent:

  1. $(p_0,p_1)$ is symmetric, i.e. $R^{\text{op}} \subseteq_{X \times X} R$.
  2. For any $(x,y): T \rightarrow X \times X$: if $(x,y) \in R$ then $(y,x) \in R$.
  3. The restriction $\tau_{X \times X} \circ (p_0,p_1)$ of the twist function $\tau_{X \times X}$ to the relation $R$ is contained in $R$, i.e. $\tau_{X \times X} \circ (p_0,p_1) \in R$ .

I am having trouble proving the step from 2 to 3, since the hypothesis doesn't immediately strike me as having anything to do with the conclusion wanted.

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    Use the definition of the twist function to rewrite 3 in a simpler form. Then it will be easier to prove what you want. – 2017-02-06
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    Thank you @Test123, that solved it form me. – 2017-02-07

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