This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem elpm

Description: The predicate "is a partial function". (Contributed by NM, 15-Nov-2007) (Revised by Mario Carneiro, 14-Nov-2013)

Ref Expression
Hypotheses elmap.1 ⊢ 𝐴 ∈ V
elmap.2 ⊢ 𝐵 ∈ V
Assertion elpm ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( Fun 𝐹 ∧ 𝐹 ⊆ ( 𝐵 × 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 elmap.1 ⊢ 𝐴 ∈ V
2 elmap.2 ⊢ 𝐵 ∈ V
3 elpmg ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( Fun 𝐹 ∧ 𝐹 ⊆ ( 𝐵 × 𝐴 ) ) ) )
4 1 2 3 mp2an ⊢ ( 𝐹 ∈ ( 𝐴 ↑pm 𝐵 ) ↔ ( Fun 𝐹 ∧ 𝐹 ⊆ ( 𝐵 × 𝐴 ) ) )