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Metamath Proof Explorer

Theorem reprss

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

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

Proof

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