Step | Hyp | Ref
| Expression |
1 | | evl1fval.o |
. . 3
⊢ 𝑂 = (eval1‘𝑅) |
2 | | fvexd 6732 |
. . . . 5
⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V) |
3 | | id 22 |
. . . . . . . . 9
⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟)) |
4 | | fveq2 6717 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) |
5 | | evl1fval.b |
. . . . . . . . . 10
⊢ 𝐵 = (Base‘𝑅) |
6 | 4, 5 | eqtr4di 2796 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
7 | 3, 6 | sylan9eqr 2800 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵) |
8 | 7 | oveq1d 7228 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m 1o) = (𝐵 ↑m
1o)) |
9 | 7, 8 | oveq12d 7231 |
. . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m
1o))) |
10 | 7 | mpteq1d 5144 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) |
11 | 10 | coeq2d 5731 |
. . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
12 | 9, 11 | mpteq12dv 5140 |
. . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))) |
13 | | simpl 486 |
. . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅) |
14 | 13 | oveq2d 7229 |
. . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅)) |
15 | | evl1fval.q |
. . . . . . 7
⊢ 𝑄 = (1o eval 𝑅) |
16 | 14, 15 | eqtr4di 2796 |
. . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄) |
17 | 12, 16 | coeq12d 5733 |
. . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o
eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)) |
18 | 2, 17 | csbied 3849 |
. . . 4
⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o
eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)) |
19 | | df-evl1 21232 |
. . . 4
⊢
eval1 = (𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o
eval 𝑟))) |
20 | | ovex 7246 |
. . . . . 6
⊢ (𝐵 ↑m (𝐵 ↑m
1o)) ∈ V |
21 | 20 | mptex 7039 |
. . . . 5
⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V |
22 | 15 | ovexi 7247 |
. . . . 5
⊢ 𝑄 ∈ V |
23 | 21, 22 | coex 7708 |
. . . 4
⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V |
24 | 18, 19, 23 | fvmpt 6818 |
. . 3
⊢ (𝑅 ∈ V →
(eval1‘𝑅)
= ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m
1o)) ↦ (𝑥
∘ (𝑦 ∈ 𝐵 ↦ (1o ×
{𝑦})))) ∘ 𝑄)) |
25 | 1, 24 | syl5eq 2790 |
. 2
⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)) |
26 | | fvprc 6709 |
. . . . 5
⊢ (¬
𝑅 ∈ V →
(eval1‘𝑅)
= ∅) |
27 | 1, 26 | syl5eq 2790 |
. . . 4
⊢ (¬
𝑅 ∈ V → 𝑂 = ∅) |
28 | | co02 6124 |
. . . 4
⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅) =
∅ |
29 | 27, 28 | eqtr4di 2796 |
. . 3
⊢ (¬
𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘
∅)) |
30 | | df-evl 21033 |
. . . . . . 7
⊢ eval =
(𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) |
31 | 30 | reldmmpo 7344 |
. . . . . 6
⊢ Rel dom
eval |
32 | 31 | ovprc2 7253 |
. . . . 5
⊢ (¬
𝑅 ∈ V →
(1o eval 𝑅) =
∅) |
33 | 15, 32 | syl5eq 2790 |
. . . 4
⊢ (¬
𝑅 ∈ V → 𝑄 = ∅) |
34 | 33 | coeq2d 5731 |
. . 3
⊢ (¬
𝑅 ∈ V → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘
∅)) |
35 | 29, 34 | eqtr4d 2780 |
. 2
⊢ (¬
𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)) |
36 | 25, 35 | pm2.61i 185 |
1
⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) |