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 ) ) )