Theorem elrn2

Description: Membership in a range. (Contributed by NM, 10-Jul-1994)

	Ref	Expression
Hypothesis	elrn.1	⊢ 𝐴 ∈ V
Assertion	elrn2	⊢ ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elrn.1	⊢ 𝐴 ∈ V
2		opeq2	⊢ ( 𝑦 = 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝐴 ⟩ )
3	2	eleq1d	⊢ ( 𝑦 = 𝐴 → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ↔ ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
4	3	exbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
5		dfrn3	⊢ ran 𝐵 = { 𝑦 ∣ ∃ 𝑥 ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 }
6	1 4 5	elab2	⊢ ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 )