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 = 𝑤 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑤 ⟩ )
2	1	eleq1d	⊢ ( 𝑦 = 𝑤 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ↔ ⟨ 𝑥 , 𝑤 ⟩ ∈ 𝐵 ) )
3	2	exbidv	⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝑤 ⟩ ∈ 𝐵 ) )
4		opeq2	⊢ ( 𝑤 = 𝐴 → ⟨ 𝑥 , 𝑤 ⟩ = ⟨ 𝑥 , 𝐴 ⟩ )
5	4	eleq1d	⊢ ( 𝑤 = 𝐴 → ( ⟨ 𝑥 , 𝑤 ⟩ ∈ 𝐵 ↔ ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
6	5	exbidv	⊢ ( 𝑤 = 𝐴 → ( ∃ 𝑥 ⟨ 𝑥 , 𝑤 ⟩ ∈ 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
7		dfrn3	⊢ ran 𝐵 = { 𝑦 ∣ ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 }
8	3 6 7	elab2gw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )