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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 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 B ↔ F : B ⟶ A
4 1 2 3 mp2an ⊢ F ∈ A B ↔ F : B ⟶ A