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Metamath Proof Explorer

Theorem fpmd

Description: A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Hypotheses fpmd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
fpmd.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
fpmd.c ⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
fpmd.f ⊢ ( 𝜑 → 𝐹 : 𝐶 ⟶ 𝐵 )
Assertion fpmd ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑pm 𝐴 ) )

Proof

Step Hyp Ref Expression
1 fpmd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2 fpmd.b ⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3 fpmd.c ⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
4 fpmd.f ⊢ ( 𝜑 → 𝐹 : 𝐶 ⟶ 𝐵 )
5 elpm2r ⊢ ( ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) ∧ ( 𝐹 : 𝐶 ⟶ 𝐵 ∧ 𝐶 ⊆ 𝐴 ) ) → 𝐹 ∈ ( 𝐵 ↑pm 𝐴 ) )
6 2 1 4 3 5 syl22anc ⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑pm 𝐴 ) )