Metamath Proof Explorer


Theorem bj-restv

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 𝑋𝑋 ) → 𝐴 ∈ ( 𝑋t 𝐴 ) )

Proof

Step Hyp Ref Expression
1 uniexr ( 𝑋𝑋𝑋 ∈ V )
2 1 adantl ( ( 𝐴 𝑋 𝑋𝑋 ) → 𝑋 ∈ V )
3 bj-restb ( 𝑋 ∈ V → ( ( 𝐴 𝑋 𝑋𝑋 ) → 𝐴 ∈ ( 𝑋t 𝐴 ) ) )
4 2 3 mpcom ( ( 𝐴 𝑋 𝑋𝑋 ) → 𝐴 ∈ ( 𝑋t 𝐴 ) )