Metamath Proof Explorer


Theorem fdmfisuppfi

Description: The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019)

Ref Expression
Hypotheses fdmfisuppfi.f ( 𝜑𝐹 : 𝐷𝑅 )
fdmfisuppfi.d ( 𝜑𝐷 ∈ Fin )
fdmfisuppfi.z ( 𝜑𝑍𝑉 )
Assertion fdmfisuppfi ( 𝜑 → ( 𝐹 supp 𝑍 ) ∈ Fin )

Proof

Step Hyp Ref Expression
1 fdmfisuppfi.f ( 𝜑𝐹 : 𝐷𝑅 )
2 fdmfisuppfi.d ( 𝜑𝐷 ∈ Fin )
3 fdmfisuppfi.z ( 𝜑𝑍𝑉 )
4 1 2 fexd ( 𝜑𝐹 ∈ V )
5 suppimacnv ( ( 𝐹 ∈ V ∧ 𝑍𝑉 ) → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
6 4 3 5 syl2anc ( 𝜑 → ( 𝐹 supp 𝑍 ) = ( 𝐹 “ ( V ∖ { 𝑍 } ) ) )
7 2 1 fisuppfi ( 𝜑 → ( 𝐹 “ ( V ∖ { 𝑍 } ) ) ∈ Fin )
8 6 7 eqeltrd ( 𝜑 → ( 𝐹 supp 𝑍 ) ∈ Fin )