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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 ) )