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 ) } ) |
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 ) } ) |