Metamath Proof Explorer

Theorem ecres

Description: Restricted coset of B . (Contributed by Peter Mazsa, 9-Dec-2018)

Ref Expression
Assertion ecres [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) = { 𝑥 ∣ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝑥 ) }

Proof

Step Hyp Ref Expression
1 elecres ⊢ ( 𝑥 ∈ V → ( 𝑥 ∈ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝑥 ) ) )
2 1 elv ⊢ ( 𝑥 ∈ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝑥 ) )
3 2 eqabi ⊢ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) = { 𝑥 ∣ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝑥 ) }