Description: The support of a function with a finite domain is always finite. (Contributed by AV, 27-Apr-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | fdmfisuppfi.f | ⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ 𝑅 ) | |
fdmfisuppfi.d | ⊢ ( 𝜑 → 𝐷 ∈ Fin ) | ||
fdmfisuppfi.z | ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) | ||
Assertion | fdmfisuppfi | ⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) ∈ Fin ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdmfisuppfi.f | ⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ 𝑅 ) | |
2 | fdmfisuppfi.d | ⊢ ( 𝜑 → 𝐷 ∈ Fin ) | |
3 | fdmfisuppfi.z | ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) | |
4 | 1 2 | fexd | ⊢ ( 𝜑 → 𝐹 ∈ V ) |
5 | suppimacnv | ⊢ ( ( 𝐹 ∈ V ∧ 𝑍 ∈ 𝑉 ) → ( 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 ) |