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Metamath Proof Explorer

Theorem eltrrelsrel

Description: For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021)

		Ref	Expression
	Assertion	eltrrelsrel	$⊢ R \in V \to (R \in TrRels \leftrightarrow TrRel R)$

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	$⊢ R \in V \to (R \in Rels \leftrightarrow Rel R)$
2	1	anbi2d	$⊢ R \in V \to ((R \circ R \subseteq R \land R \in Rels) \leftrightarrow (R \circ R \subseteq R \land Rel R))$
3		eltrrels2	$⊢ R \in TrRels \leftrightarrow (R \circ R \subseteq R \land R \in Rels)$
4		dftrrel2	$⊢ TrRel R \leftrightarrow (R \circ R \subseteq R \land Rel R)$
5	2 3 4	3bitr4g	$⊢ R \in V \to (R \in TrRels \leftrightarrow TrRel R)$