Metamath Proof Explorer


Theorem frnnn0fsuppg

Description: Version of frnnn0fsupp avoiding ax-rep by assuming F is a set rather than its domain I . (Contributed by SN, 5-Aug-2024)

Ref Expression
Assertion frnnn0fsuppg ( ( 𝐹𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ℕ ) ∈ Fin ) )

Proof

Step Hyp Ref Expression
1 ffun ( 𝐹 : 𝐼 ⟶ ℕ0 → Fun 𝐹 )
2 simpl ( ( 𝐹𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → 𝐹𝑉 )
3 c0ex 0 ∈ V
4 funisfsupp ( ( Fun 𝐹𝐹𝑉 ∧ 0 ∈ V ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) )
5 3 4 mp3an3 ( ( Fun 𝐹𝐹𝑉 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) )
6 1 2 5 syl2an2 ( ( 𝐹𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) )
7 frnnn0suppg ( ( 𝐹𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 supp 0 ) = ( 𝐹 “ ℕ ) )
8 7 eleq1d ( ( 𝐹𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( ( 𝐹 supp 0 ) ∈ Fin ↔ ( 𝐹 “ ℕ ) ∈ Fin ) )
9 6 8 bitrd ( ( 𝐹𝑉𝐹 : 𝐼 ⟶ ℕ0 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 “ ℕ ) ∈ Fin ) )