Metamath Proof Explorer


Theorem cleq1

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

Ref Expression
Assertion cleq1 ( 𝑅 = 𝑆 { 𝑟 ∣ ( 𝑅𝑟𝜑 ) } = { 𝑟 ∣ ( 𝑆𝑟𝜑 ) } )

Proof

Step Hyp Ref Expression
1 cleq1lem ( 𝑅 = 𝑆 → ( ( 𝑅𝑟𝜑 ) ↔ ( 𝑆𝑟𝜑 ) ) )
2 1 abbidv ( 𝑅 = 𝑆 → { 𝑟 ∣ ( 𝑅𝑟𝜑 ) } = { 𝑟 ∣ ( 𝑆𝑟𝜑 ) } )
3 2 inteqd ( 𝑅 = 𝑆 { 𝑟 ∣ ( 𝑅𝑟𝜑 ) } = { 𝑟 ∣ ( 𝑆𝑟𝜑 ) } )