Metamath Proof Explorer


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 ( 𝑅𝑉 { 𝑟 ∣ ( 𝑅𝑟 ∧ ( 𝑟𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 trclublem ( 𝑅𝑉 → ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) ∈ { 𝑟 ∣ ( 𝑅𝑟 ∧ ( 𝑟𝑟 ) ⊆ 𝑟 ) } )
2 intss1 ( ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) ∈ { 𝑟 ∣ ( 𝑅𝑟 ∧ ( 𝑟𝑟 ) ⊆ 𝑟 ) } → { 𝑟 ∣ ( 𝑅𝑟 ∧ ( 𝑟𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )
3 1 2 syl ( 𝑅𝑉 { 𝑟 ∣ ( 𝑅𝑟 ∧ ( 𝑟𝑟 ) ⊆ 𝑟 ) } ⊆ ( 𝑅 ∪ ( dom 𝑅 × ran 𝑅 ) ) )