Metamath Proof Explorer


Theorem trclfvlb2

Description: The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020)

Ref Expression
Assertion trclfvlb2 ( 𝑅𝑉 → ( 𝑅𝑅 ) ⊆ ( t+ ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 trclfvcotr ( 𝑅𝑉 → ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) )
2 trclfvlb ( 𝑅𝑉𝑅 ⊆ ( t+ ‘ 𝑅 ) )
3 1 2 2 trrelssd ( 𝑅𝑉 → ( 𝑅𝑅 ) ⊆ ( t+ ‘ 𝑅 ) )