Theorem dftrrels3

Description: Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021)

	Ref	Expression
Assertion	dftrrels3	$$⊢\mathrm{TrRels}=\left\{r\in \mathrm{Rels}|\forall x\forall y\forall z\left(\left(xry\wedge yrz\right)\to xrz\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dftrrels2	$$⊢\mathrm{TrRels}=\left\{r\in \mathrm{Rels}|r\circ r\subseteq r\right\}$$
2		cotr	$$⊢r\circ r\subseteq r\leftrightarrow \forall x\forall y\forall z\left(\left(xry\wedge yrz\right)\to xrz\right)$$
3	1 2	rabbieq	$$⊢\mathrm{TrRels}=\left\{r\in \mathrm{Rels}|\forall x\forall y\forall z\left(\left(xry\wedge yrz\right)\to xrz\right)\right\}$$