| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-of 7698 | . . . 4
⊢ 
∘f 𝑅 =
(𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | 
| 2 | 1 | a1i 11 | . . 3
⊢ (𝐹 ∈ V →
∘f 𝑅 =
(𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))) | 
| 3 |  | dmeq 5913 | . . . . . . 7
⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | 
| 4 |  | dmeq 5913 | . . . . . . 7
⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅) | 
| 5 | 3, 4 | ineqan12d 4221 | . . . . . 6
⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅)) | 
| 6 | 5 | mpteq1d 5236 | . . . . 5
⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | 
| 7 | 6 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | 
| 8 |  | dm0 5930 | . . . . . . . 8
⊢ dom
∅ = ∅ | 
| 9 | 8 | ineq2i 4216 | . . . . . . 7
⊢ (dom
𝐹 ∩ dom ∅) = (dom
𝐹 ∩
∅) | 
| 10 |  | in0 4394 | . . . . . . 7
⊢ (dom
𝐹 ∩ ∅) =
∅ | 
| 11 | 9, 10 | eqtri 2764 | . . . . . 6
⊢ (dom
𝐹 ∩ dom ∅) =
∅ | 
| 12 | 11 | a1i 11 | . . . . 5
⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) =
∅) | 
| 13 | 12 | mpteq1d 5236 | . . . 4
⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | 
| 14 |  | mpt0 6709 | . . . . 5
⊢ (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅ | 
| 15 | 14 | a1i 11 | . . . 4
⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅) | 
| 16 | 7, 13, 15 | 3eqtrd 2780 | . . 3
⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅) | 
| 17 |  | id 22 | . . 3
⊢ (𝐹 ∈ V → 𝐹 ∈ V) | 
| 18 |  | 0ex 5306 | . . . 4
⊢ ∅
∈ V | 
| 19 | 18 | a1i 11 | . . 3
⊢ (𝐹 ∈ V → ∅ ∈
V) | 
| 20 | 2, 16, 17, 19, 19 | ovmpod 7586 | . 2
⊢ (𝐹 ∈ V → (𝐹 ∘f 𝑅∅) =
∅) | 
| 21 | 1 | reldmmpo 7568 | . . 3
⊢ Rel dom
∘f 𝑅 | 
| 22 | 21 | ovprc1 7471 | . 2
⊢ (¬
𝐹 ∈ V → (𝐹 ∘f 𝑅∅) =
∅) | 
| 23 | 20, 22 | pm2.61i 182 | 1
⊢ (𝐹 ∘f 𝑅∅) =
∅ |