Theorem elrn

Description: Membership in a range. (Contributed by NM, 2-Apr-2004)

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

Proof

Step	Hyp	Ref	Expression
1		elrn.1	⊢ 𝐴 ∈ V
2		elrng	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 𝑥 𝐵 𝐴 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 𝑥 𝐵 𝐴 )