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	1	elrn2	⊢ ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 )
3		df-br	⊢ ( 𝑥 𝐵 𝐴 ↔ ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 )
4	3	exbii	⊢ ( ∃ 𝑥 𝑥 𝐵 𝐴 ↔ ∃ 𝑥 ⟨ 𝑥 , 𝐴 ⟩ ∈ 𝐵 )
5	2 4	bitr4i	⊢ ( 𝐴 ∈ ran 𝐵 ↔ ∃ 𝑥 𝑥 𝐵 𝐴 )