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 ∈ 𝑉 → ( 𝑅 ∈ TrRels ↔ TrRel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		elrelsrel	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ Rels ↔ Rel 𝑅 ) )
2	1	anbi2d	⊢ ( 𝑅 ∈ 𝑉 → ( ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ Rel 𝑅 ) ) )
3		eltrrels2	⊢ ( 𝑅 ∈ TrRels ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )
4		dftrrel2	⊢ ( TrRel 𝑅 ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ Rel 𝑅 ) )
5	2 3 4	3bitr4g	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∈ TrRels ↔ TrRel 𝑅 ) )