Theorem elmap

Description: Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003)

	Ref	Expression
Hypotheses	elmap.1	⊢ 𝐴 ∈ V
	elmap.2	⊢ 𝐵 ∈ V
Assertion	elmap	⊢ ( 𝐹 ∈ ( 𝐴 ↑m 𝐵 ) ↔ 𝐹 : 𝐵 ⟶ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elmap.1	⊢ 𝐴 ∈ V
2		elmap.2	⊢ 𝐵 ∈ V
3		elmapg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐹 ∈ ( 𝐴 ↑m 𝐵 ) ↔ 𝐹 : 𝐵 ⟶ 𝐴 ) )
4	1 2 3	mp2an	⊢ ( 𝐹 ∈ ( 𝐴 ↑m 𝐵 ) ↔ 𝐹 : 𝐵 ⟶ 𝐴 )