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Metamath Proof Explorer

Theorem fsuppmptdm

Description: A mapping with a finite domain is finitely supported. (Contributed by AV, 7-Jun-2019)

Ref Expression
Hypotheses fsuppmptdm.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝑌 )
fsuppmptdm.a ⊢ ( 𝜑 → 𝐴 ∈ Fin )
fsuppmptdm.y ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑌 ∈ 𝑉 )
fsuppmptdm.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑊 )
Assertion fsuppmptdm ( 𝜑 → 𝐹 finSupp 𝑍 )

Proof

Step Hyp Ref Expression
1 fsuppmptdm.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝑌 )
2 fsuppmptdm.a ⊢ ( 𝜑 → 𝐴 ∈ Fin )
3 fsuppmptdm.y ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑌 ∈ 𝑉 )
4 fsuppmptdm.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑊 )
5 3 1 fmptd ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑉 )
6 5 2 4 fdmfifsupp ⊢ ( 𝜑 → 𝐹 finSupp 𝑍 )