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 }