Theorem fpm

Description: A total function 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	fpm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ∈ ( 𝐵 ↑pm 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		elmap.1	⊢ 𝐴 ∈ V
2		elmap.2	⊢ 𝐵 ∈ V
3		fpmg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐹 ∈ ( 𝐵 ↑pm 𝐴 ) )
4	1 2 3	mp3an12	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ∈ ( 𝐵 ↑pm 𝐴 ) )