Step |
Hyp |
Ref |
Expression |
1 |
|
df-seq |
⊢ seq 𝑀 ( + , 𝐹 ) = ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) “ ω ) |
2 |
|
rdgfun |
⊢ Fun rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) |
3 |
|
omex |
⊢ ω ∈ V |
4 |
|
funimaexg |
⊢ ( ( Fun rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) ∧ ω ∈ V ) → ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) “ ω ) ∈ V ) |
5 |
2 3 4
|
mp2an |
⊢ ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) “ ω ) ∈ V |
6 |
1 5
|
eqeltri |
⊢ seq 𝑀 ( + , 𝐹 ) ∈ V |