| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | evl1fval.o | . . 3
⊢ 𝑂 = (eval1‘𝑅) | 
| 2 |  | fvexd 6921 | . . . . 5
⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V) | 
| 3 |  | id 22 | . . . . . . . . 9
⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟)) | 
| 4 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | 
| 5 |  | evl1fval.b | . . . . . . . . . 10
⊢ 𝐵 = (Base‘𝑅) | 
| 6 | 4, 5 | eqtr4di 2795 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) | 
| 7 | 3, 6 | sylan9eqr 2799 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵) | 
| 8 | 7 | oveq1d 7446 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m 1o) = (𝐵 ↑m
1o)) | 
| 9 | 7, 8 | oveq12d 7449 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m
1o))) | 
| 10 | 7 | mpteq1d 5237 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) | 
| 11 | 10 | coeq2d 5873 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) | 
| 12 | 9, 11 | mpteq12dv 5233 | . . . . . 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 482 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅) | 
| 14 | 13 | oveq2d 7447 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅)) | 
| 15 |  | evl1fval.q | . . . . . . 7
⊢ 𝑄 = (1o eval 𝑅) | 
| 16 | 14, 15 | eqtr4di 2795 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄) | 
| 17 | 12, 16 | coeq12d 5875 | . . . . 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 3935 | . . . 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 22320 | . . . 4
⊢
eval1 = (𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o
eval 𝑟))) | 
| 20 |  | ovex 7464 | . . . . . 6
⊢ (𝐵 ↑m (𝐵 ↑m
1o)) ∈ V | 
| 21 | 20 | mptex 7243 | . . . . 5
⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V | 
| 22 | 15 | ovexi 7465 | . . . . 5
⊢ 𝑄 ∈ V | 
| 23 | 21, 22 | coex 7952 | . . . 4
⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V | 
| 24 | 18, 19, 23 | fvmpt 7016 | . . 3
⊢ (𝑅 ∈ V →
(eval1‘𝑅)
= ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m
1o)) ↦ (𝑥
∘ (𝑦 ∈ 𝐵 ↦ (1o ×
{𝑦})))) ∘ 𝑄)) | 
| 25 | 1, 24 | eqtrid 2789 | . 2
⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)) | 
| 26 |  | fvprc 6898 | . . . . 5
⊢ (¬
𝑅 ∈ V →
(eval1‘𝑅)
= ∅) | 
| 27 | 1, 26 | eqtrid 2789 | . . . 4
⊢ (¬
𝑅 ∈ V → 𝑂 = ∅) | 
| 28 |  | co02 6280 | . . . 4
⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅) =
∅ | 
| 29 | 27, 28 | eqtr4di 2795 | . . 3
⊢ (¬
𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘
∅)) | 
| 30 |  | df-evl 22099 | . . . . . . 7
⊢  eval =
(𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) | 
| 31 | 30 | reldmmpo 7567 | . . . . . 6
⊢ Rel dom
eval | 
| 32 | 31 | ovprc2 7471 | . . . . 5
⊢ (¬
𝑅 ∈ V →
(1o eval 𝑅) =
∅) | 
| 33 | 15, 32 | eqtrid 2789 | . . . 4
⊢ (¬
𝑅 ∈ V → 𝑄 = ∅) | 
| 34 | 33 | coeq2d 5873 | . . 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 182 | 1
⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦
(𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) |