Description: The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006)
Ref | Expression | ||
---|---|---|---|
Assertion | relin2 | ⊢ ( Rel 𝐵 → Rel ( 𝐴 ∩ 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 | ⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵 | |
2 | relss | ⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵 → ( Rel 𝐵 → Rel ( 𝐴 ∩ 𝐵 ) ) ) | |
3 | 1 2 | ax-mp | ⊢ ( Rel 𝐵 → Rel ( 𝐴 ∩ 𝐵 ) ) |