Metamath Proof Explorer


Theorem dprdffsupp

Description: A finitely supported function in S is a finitely supported function. (Contributed by Mario Carneiro, 25-Apr-2016) (Revised by AV, 11-Jul-2019)

Ref Expression
Hypotheses dprdff.w 𝑊 = { X 𝑖𝐼 ( 𝑆𝑖 ) ∣ finSupp 0 }
dprdff.1 ( 𝜑𝐺 dom DProd 𝑆 )
dprdff.2 ( 𝜑 → dom 𝑆 = 𝐼 )
dprdff.3 ( 𝜑𝐹𝑊 )
Assertion dprdffsupp ( 𝜑𝐹 finSupp 0 )

Proof

Step Hyp Ref Expression
1 dprdff.w 𝑊 = { X 𝑖𝐼 ( 𝑆𝑖 ) ∣ finSupp 0 }
2 dprdff.1 ( 𝜑𝐺 dom DProd 𝑆 )
3 dprdff.2 ( 𝜑 → dom 𝑆 = 𝐼 )
4 dprdff.3 ( 𝜑𝐹𝑊 )
5 1 2 3 dprdw ( 𝜑 → ( 𝐹𝑊 ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) ∧ 𝐹 finSupp 0 ) ) )
6 4 5 mpbid ( 𝜑 → ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) ∧ 𝐹 finSupp 0 ) )
7 6 simp3d ( 𝜑𝐹 finSupp 0 )