Theorem eltrrels2

Description: Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021)

	Ref	Expression
Assertion	eltrrels2	⊢ ( 𝑅 ∈ TrRels ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )

Proof

Step	Hyp	Ref	Expression
1		dftrrels2	⊢ TrRels = { 𝑟 ∈ Rels ∣ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 }
2		coideq	⊢ ( 𝑟 = 𝑅 → ( 𝑟 ∘ 𝑟 ) = ( 𝑅 ∘ 𝑅 ) )
3		id	⊢ ( 𝑟 = 𝑅 → 𝑟 = 𝑅 )
4	2 3	sseq12d	⊢ ( 𝑟 = 𝑅 → ( ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ↔ ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ) )
5	1 4	rabeqel	⊢ ( 𝑅 ∈ TrRels ↔ ( ( 𝑅 ∘ 𝑅 ) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ) )