Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022)
Ref | Expression | ||
---|---|---|---|
Assertion | brres | ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelres | ⊢ ( 𝐶 ∈ 𝑉 → ( 〈 𝐵 , 𝐶 〉 ∈ ( 𝑅 ↾ 𝐴 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 〈 𝐵 , 𝐶 〉 ∈ 𝑅 ) ) ) | |
2 | df-br | ⊢ ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ 〈 𝐵 , 𝐶 〉 ∈ ( 𝑅 ↾ 𝐴 ) ) | |
3 | df-br | ⊢ ( 𝐵 𝑅 𝐶 ↔ 〈 𝐵 , 𝐶 〉 ∈ 𝑅 ) | |
4 | 3 | anbi2i | ⊢ ( ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 〈 𝐵 , 𝐶 〉 ∈ 𝑅 ) ) |
5 | 1 2 4 | 3bitr4g | ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) ) |