| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | bj-brresdm 37147 | . . . 4
⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐴 ∈ 𝐶) | 
| 2 |  | relres 6023 | . . . . 5
⊢ Rel ( I
↾ 𝐶) | 
| 3 | 2 | brrelex2i 5742 | . . . 4
⊢ (𝐴( I ↾ 𝐶)𝐵 → 𝐵 ∈ V) | 
| 4 | 1, 3 | jca 511 | . . 3
⊢ (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V)) | 
| 5 | 4 | adantl 481 | . 2
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴( I ↾ 𝐶)𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V)) | 
| 6 |  | eqimss 4042 | . . . . . 6
⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | 
| 7 |  | dfss2 3969 | . . . . . 6
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | 
| 8 | 6, 7 | sylib 218 | . . . . 5
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) | 
| 9 | 8 | adantl 481 | . . . 4
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐴) | 
| 10 |  | simpl 482 | . . . 4
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐶) | 
| 11 | 9, 10 | eqeltrrd 2842 | . . 3
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐶) | 
| 12 |  | eqimss2 4043 | . . . . . . 7
⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | 
| 13 |  | sseqin2 4223 | . . . . . . 7
⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | 
| 14 | 12, 13 | sylib 218 | . . . . . 6
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵) | 
| 15 | 14 | adantl 481 | . . . . 5
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∩ 𝐵) = 𝐵) | 
| 16 | 15, 10 | eqeltrrd 2842 | . . . 4
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐶) | 
| 17 | 16 | elexd 3504 | . . 3
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐵 ∈ V) | 
| 18 | 11, 17 | jca 511 | . 2
⊢ (((𝐴 ∩ 𝐵) ∈ 𝐶 ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V)) | 
| 19 |  | brres 6004 | . . . 4
⊢ (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵))) | 
| 20 | 19 | adantl 481 | . . 3
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵))) | 
| 21 |  | eqeq12 2754 | . . . . 5
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 = 𝑦 ↔ 𝐴 = 𝐵)) | 
| 22 |  | df-id 5578 | . . . . 5
⊢  I =
{〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | 
| 23 | 21, 22 | brabga 5539 | . . . 4
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | 
| 24 | 23 | anbi2d 630 | . . 3
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))) | 
| 25 |  | simp3 1139 | . . . . 5
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | 
| 26 | 25 | 3expib 1123 | . . . 4
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)) | 
| 27 |  | 3simpb 1150 | . . . . 5
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V ∧ 𝐴 = 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵)) | 
| 28 | 27 | 3expia 1122 | . . . 4
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))) | 
| 29 | 26, 28 | impbid 212 | . . 3
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → ((𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵) ↔ 𝐴 = 𝐵)) | 
| 30 | 20, 24, 29 | 3bitrd 305 | . 2
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵)) | 
| 31 | 5, 18, 30 | pm5.21nd 802 | 1
⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵)) |