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		elrn2g	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 )