Metamath Proof Explorer


Theorem reprss

Description: Representations with terms in a subset. (Contributed by Thierry Arnoux, 11-Dec-2021)

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

Proof

Step Hyp Ref Expression
1 reprval.a ( 𝜑𝐴 ⊆ ℕ )
2 reprval.m ( 𝜑𝑀 ∈ ℤ )
3 reprval.s ( 𝜑𝑆 ∈ ℕ0 )
4 reprss.1 ( 𝜑𝐵𝐴 )
5 nnex ℕ ∈ V
6 5 a1i ( 𝜑 → ℕ ∈ V )
7 6 1 ssexd ( 𝜑𝐴 ∈ V )
8 mapss ( ( 𝐴 ∈ V ∧ 𝐵𝐴 ) → ( 𝐵m ( 0 ..^ 𝑆 ) ) ⊆ ( 𝐴m ( 0 ..^ 𝑆 ) ) )
9 7 4 8 syl2anc ( 𝜑 → ( 𝐵m ( 0 ..^ 𝑆 ) ) ⊆ ( 𝐴m ( 0 ..^ 𝑆 ) ) )
10 9 sselda ( ( 𝜑𝑐 ∈ ( 𝐵m ( 0 ..^ 𝑆 ) ) ) → 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) )
11 10 adantrr ( ( 𝜑 ∧ ( 𝑐 ∈ ( 𝐵m ( 0 ..^ 𝑆 ) ) ∧ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 ) ) → 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) )
12 11 rabss3d ( 𝜑 → { 𝑐 ∈ ( 𝐵m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } ⊆ { 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } )
13 4 1 sstrd ( 𝜑𝐵 ⊆ ℕ )
14 13 2 3 reprval ( 𝜑 → ( 𝐵 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐵m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } )
15 1 2 3 reprval ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = { 𝑐 ∈ ( 𝐴m ( 0 ..^ 𝑆 ) ) ∣ Σ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( 𝑐𝑎 ) = 𝑀 } )
16 12 14 15 3sstr4d ( 𝜑 → ( 𝐵 ( repr ‘ 𝑆 ) 𝑀 ) ⊆ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) )