Theorem trclubg

Description: The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020)

	Ref	Expression
Assertion	trclubg	⊢ ( 𝑅 ∈ 𝑉 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		trclublem	⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) ∈ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )
2		intss1	⊢ ( ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) ∈ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )
3	1 2	syl	⊢ ( 𝑅 ∈ 𝑉 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )