Theorem trclfvub

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

	Ref	Expression
Assertion	trclfvub	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		trclfv	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) = ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
2		trclubg	⊢ ( 𝑅 ∈ 𝑉 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )
3	1 2	eqsstrd	⊢ ( 𝑅 ∈ 𝑉 → ( t+ ‘ 𝑅 ) ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )