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 ) ) ) |
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 ) ) ) |