Theorem trcleq1

Description: Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020)

	Ref	Expression
Assertion	trcleq1	⊢ ( 𝑅 = 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } = ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )

Proof

Step	Hyp	Ref	Expression
1		cleq1	⊢ ( 𝑅 = 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } = ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ ( 𝑟 ∘ 𝑟 ) ⊆ 𝑟 ) } )