Metamath Proof Explorer


Theorem frnnn0fsupp

Description: A function on NN0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019)

Ref Expression
Assertion frnnn0fsupp ( ( 𝐼𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ℕ ) ∈ Fin ) )

Proof

Step Hyp Ref Expression
1 c0ex 0 ∈ V
2 frnfsuppbi ( ( 𝐼𝑉 ∧ 0 ∈ V ) → ( 𝐹 : 𝐼 ⟶ ℕ0 → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ( ℕ0 ∖ { 0 } ) ) ∈ Fin ) ) )
3 1 2 mpan2 ( 𝐼𝑉 → ( 𝐹 : 𝐼 ⟶ ℕ0 → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ( ℕ0 ∖ { 0 } ) ) ∈ Fin ) ) )
4 3 imp ( ( 𝐼𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ( ℕ0 ∖ { 0 } ) ) ∈ Fin ) )
5 dfn2 ℕ = ( ℕ0 ∖ { 0 } )
6 5 imaeq2i ( 𝐹 “ ℕ ) = ( 𝐹 “ ( ℕ0 ∖ { 0 } ) )
7 6 eleq1i ( ( 𝐹 “ ℕ ) ∈ Fin ↔ ( 𝐹 “ ( ℕ0 ∖ { 0 } ) ) ∈ Fin )
8 4 7 bitr4di ( ( 𝐼𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ℕ ) ∈ Fin ) )