Theorem trclfvlb

Description: The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020)

	Ref	Expression
Assertion	trclfvlb	⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ⊆ ( t+ ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		ssmin	⊢ 𝑅 ⊆ ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) }
2		trclfv	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) = ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
3	1 2	sseqtrrid	⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ⊆ ( t+ ‘ 𝑅 ) )