Theorem cleq1

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

	Ref	Expression
Assertion	cleq1	⊢ ( 𝑅 = 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } = ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } )

Proof

Step	Hyp	Ref	Expression
1		cleq1lem	⊢ ( 𝑅 = 𝑆 → ( ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) ↔ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) ) )
2	1	abbidv	⊢ ( 𝑅 = 𝑆 → { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } = { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } )
3	2	inteqd	⊢ ( 𝑅 = 𝑆 → ∩ { 𝑟 ∣ ( 𝑅 ⊆ 𝑟 ∧ 𝜑 ) } = ∩ { 𝑟 ∣ ( 𝑆 ⊆ 𝑟 ∧ 𝜑 ) } )