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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 ∣ ( 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 ) } )