Step | Hyp | Ref
| Expression |
1 | | 0nn0 12178 |
. . . . 5
⊢ 0 ∈
ℕ0 |
2 | | iftrue 4462 |
. . . . . 6
⊢ (𝑛 = 0 → if(𝑛 = 0, ∅, (𝑛 − 1)) =
∅) |
3 | | eqid 2738 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 − 1))) = (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))) |
4 | | 0ex 5226 |
. . . . . 6
⊢ ∅
∈ V |
5 | 2, 3, 4 | fvmpt 6857 |
. . . . 5
⊢ (0 ∈
ℕ0 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) =
∅) |
6 | 1, 5 | mp1i 13 |
. . . 4
⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) =
∅) |
7 | | 0elpw 5273 |
. . . 4
⊢ ∅
∈ 𝒫 ℕ0 |
8 | 6, 7 | eqeltrdi 2847 |
. . 3
⊢ (𝜑 → ((𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘0) ∈
𝒫 ℕ0) |
9 | | df-ov 7258 |
. . . . 5
⊢ (𝑥(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑦) = ((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))‘〈𝑥, 𝑦〉) |
10 | | elpwi 4539 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ 𝒫
ℕ0 → 𝑝 ⊆
ℕ0) |
11 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑝 ∈ 𝒫
ℕ0 ∧ 𝑚
∈ ℕ0) → 𝑝 ⊆
ℕ0) |
12 | | ssrab2 4009 |
. . . . . . . . . 10
⊢ {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} ⊆
ℕ0 |
13 | | sadcl 16097 |
. . . . . . . . . 10
⊢ ((𝑝 ⊆ ℕ0
∧ {𝑛 ∈
ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)} ⊆ ℕ0) →
(𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ⊆
ℕ0) |
14 | 11, 12, 13 | sylancl 585 |
. . . . . . . . 9
⊢ ((𝑝 ∈ 𝒫
ℕ0 ∧ 𝑚
∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ⊆
ℕ0) |
15 | | nn0ex 12169 |
. . . . . . . . . 10
⊢
ℕ0 ∈ V |
16 | 15 | elpw2 5264 |
. . . . . . . . 9
⊢ ((𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0
↔ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ⊆
ℕ0) |
17 | 14, 16 | sylibr 233 |
. . . . . . . 8
⊢ ((𝑝 ∈ 𝒫
ℕ0 ∧ 𝑚
∈ ℕ0) → (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫
ℕ0) |
18 | 17 | rgen2 3126 |
. . . . . . 7
⊢
∀𝑝 ∈
𝒫 ℕ0∀𝑚 ∈ ℕ0 (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫
ℕ0 |
19 | | eqid 2738 |
. . . . . . . 8
⊢ (𝑝 ∈ 𝒫
ℕ0, 𝑚
∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})) = (𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})) |
20 | 19 | fmpo 7881 |
. . . . . . 7
⊢
(∀𝑝 ∈
𝒫 ℕ0∀𝑚 ∈ ℕ0 (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}) ∈ 𝒫 ℕ0
↔ (𝑝 ∈ 𝒫
ℕ0, 𝑚
∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})):(𝒫 ℕ0 ×
ℕ0)⟶𝒫 ℕ0) |
21 | 18, 20 | mpbi 229 |
. . . . . 6
⊢ (𝑝 ∈ 𝒫
ℕ0, 𝑚
∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})):(𝒫 ℕ0 ×
ℕ0)⟶𝒫 ℕ0 |
22 | 21, 7 | f0cli 6956 |
. . . . 5
⊢ ((𝑝 ∈ 𝒫
ℕ0, 𝑚
∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))‘〈𝑥, 𝑦〉) ∈ 𝒫
ℕ0 |
23 | 9, 22 | eqeltri 2835 |
. . . 4
⊢ (𝑥(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑦) ∈ 𝒫
ℕ0 |
24 | 23 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝒫 ℕ0 ∧
𝑦 ∈ V)) → (𝑥(𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)}))𝑦) ∈ 𝒫
ℕ0) |
25 | | nn0uz 12549 |
. . 3
⊢
ℕ0 = (ℤ≥‘0) |
26 | | 0zd 12261 |
. . 3
⊢ (𝜑 → 0 ∈
ℤ) |
27 | | fvexd 6771 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(0 +
1))) → ((𝑛 ∈
ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))‘𝑥) ∈ V) |
28 | 8, 24, 25, 26, 27 | seqf2 13670 |
. 2
⊢ (𝜑 → seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 −
1)))):ℕ0⟶𝒫
ℕ0) |
29 | | smuval.p |
. . 3
⊢ 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) |
30 | 29 | feq1i 6575 |
. 2
⊢ (𝑃:ℕ0⟶𝒫
ℕ0 ↔ seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0
↦ (𝑝 sadd {𝑛 ∈ ℕ0
∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 −
1)))):ℕ0⟶𝒫
ℕ0) |
31 | 28, 30 | sylibr 233 |
1
⊢ (𝜑 → 𝑃:ℕ0⟶𝒫
ℕ0) |