Description: Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | brssrres | ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( S ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brres | ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( S ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 S 𝐶 ) ) ) | |
2 | brssr | ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 S 𝐶 ↔ 𝐵 ⊆ 𝐶 ) ) | |
3 | 2 | anbi2d | ⊢ ( 𝐶 ∈ 𝑉 → ( ( 𝐵 ∈ 𝐴 ∧ 𝐵 S 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶 ) ) ) |
4 | 1 3 | bitrd | ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( S ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶 ) ) ) |