Theorem elrn2g

Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011)

	Ref	Expression
Assertion	elrn2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		opeq2	⊢ ( 𝑦 = 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝐴 ⟩ )
2	1	eleq1d	⊢ ( 𝑦 = 𝐴 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ↔ ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
3	2	exbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
4		dfrn3	⊢ ran 𝐵 = { 𝑦 ∣ ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 }
5	3 4	elab2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )