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Metamath Proof Explorer

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 )