Metamath Proof Explorer


Theorem reprfi

Description: Bounded representations are finite sets. (Contributed by Thierry Arnoux, 7-Dec-2021)

Ref Expression
Hypotheses reprval.a ( 𝜑𝐴 ⊆ ℕ )
reprval.m ( 𝜑𝑀 ∈ ℤ )
reprval.s ( 𝜑𝑆 ∈ ℕ0 )
reprfi.1 ( 𝜑𝐴 ∈ Fin )
Assertion reprfi ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ∈ Fin )

Proof

Step Hyp Ref Expression
1 reprval.a ( 𝜑𝐴 ⊆ ℕ )
2 reprval.m ( 𝜑𝑀 ∈ ℤ )
3 reprval.s ( 𝜑𝑆 ∈ ℕ0 )
4 reprfi.1 ( 𝜑𝐴 ∈ Fin )
5 1 2 3 reprval ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } )
6 fzofi ( 0 ..^ 𝑆 ) ∈ Fin
7 mapfi ( ( 𝐴 ∈ Fin ∧ ( 0 ..^ 𝑆 ) ∈ Fin ) → ( 𝐴m ( 0 ..^ 𝑆 ) ) ∈ Fin )
8 4 6 7 sylancl ( 𝜑 → ( 𝐴m ( 0 ..^ 𝑆 ) ) ∈ Fin )
9 rabfi ( ( 𝐴m ( 0 ..^ 𝑆 ) ) ∈ Fin → { 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } ∈ Fin )
10 8 9 syl ( 𝜑 → { 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } ∈ Fin )
11 5 10 eqeltrd ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ∈ Fin )