Metamath Proof Explorer


Theorem mptv

Description: Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013)

Ref Expression
Assertion mptv ( 𝑥 ∈ V ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 = 𝐵 }

Proof

Step Hyp Ref Expression
1 df-mpt ( 𝑥 ∈ V ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ V ∧ 𝑦 = 𝐵 ) }
2 vex 𝑥 ∈ V
3 2 biantrur ( 𝑦 = 𝐵 ↔ ( 𝑥 ∈ V ∧ 𝑦 = 𝐵 ) )
4 3 opabbii { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 = 𝐵 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ V ∧ 𝑦 = 𝐵 ) }
5 1 4 eqtr4i ( 𝑥 ∈ V ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑦 = 𝐵 }