Metamath Proof Explorer

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