Metamath Proof Explorer

Theorem elpm2

Description: The predicate "is a partial function". (Contributed by NM, 15-Nov-2007) (Revised by Mario Carneiro, 31-Dec-2013)

Ref Expression
Hypotheses elmap.1 ⊢ A ∈ V
elmap.2 ⊢ B ∈ V
Assertion elpm2 ⊢ F ∈ A ↑ 𝑝𝑚 B ↔ F : dom ⁡ F ⟶ A ∧ dom ⁡ F ⊆ B

Proof

Step Hyp Ref Expression
1 elmap.1 ⊢ A ∈ V
2 elmap.2 ⊢ B ∈ V
3 elpm2g ⊢ A ∈ V ∧ B ∈ V → F ∈ A ↑ 𝑝𝑚 B ↔ F : dom ⁡ F ⟶ A ∧ dom ⁡ F ⊆ B
4 1 2 3 mp2an ⊢ F ∈ A ↑ 𝑝𝑚 B ↔ F : dom ⁡ F ⟶ A ∧ dom ⁡ F ⊆ B