Description: An elementwise intersection by a subset on a family containing the whole set contains the whole subset. (Contributed by BJ, 27-Apr-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | bj-restv | ⊢ ( ( 𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ 𝑋 ) → 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexr | ⊢ ( ∪ 𝑋 ∈ 𝑋 → 𝑋 ∈ V ) | |
2 | 1 | adantl | ⊢ ( ( 𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ 𝑋 ) → 𝑋 ∈ V ) |
3 | bj-restb | ⊢ ( 𝑋 ∈ V → ( ( 𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ 𝑋 ) → 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ) ) | |
4 | 2 3 | mpcom | ⊢ ( ( 𝐴 ⊆ ∪ 𝑋 ∧ ∪ 𝑋 ∈ 𝑋 ) → 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ) |