Description: The empty set is a finitely supported function. (Contributed by AV, 19-Jul-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | 0fsupp | ⊢ ( 𝑍 ∈ 𝑉 → ∅ finSupp 𝑍 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supp0 | ⊢ ( 𝑍 ∈ 𝑉 → ( ∅ supp 𝑍 ) = ∅ ) | |
2 | 0fin | ⊢ ∅ ∈ Fin | |
3 | 1 2 | eqeltrdi | ⊢ ( 𝑍 ∈ 𝑉 → ( ∅ supp 𝑍 ) ∈ Fin ) |
4 | fun0 | ⊢ Fun ∅ | |
5 | 0ex | ⊢ ∅ ∈ V | |
6 | funisfsupp | ⊢ ( ( Fun ∅ ∧ ∅ ∈ V ∧ 𝑍 ∈ 𝑉 ) → ( ∅ finSupp 𝑍 ↔ ( ∅ supp 𝑍 ) ∈ Fin ) ) | |
7 | 4 5 6 | mp3an12 | ⊢ ( 𝑍 ∈ 𝑉 → ( ∅ finSupp 𝑍 ↔ ( ∅ supp 𝑍 ) ∈ Fin ) ) |
8 | 3 7 | mpbird | ⊢ ( 𝑍 ∈ 𝑉 → ∅ finSupp 𝑍 ) |