| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 1st2nd2 8054 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) | 
| 2 | 1 | ad2antll 729 | . . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) | 
| 3 |  | simprl 770 | . . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 ∈ 𝐴) | 
| 4 | 2, 3 | eqeltrrd 2841 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∈
𝐴) | 
| 5 |  | fvex 6918 | . . . . . . 7
⊢
(1st ‘𝑥) ∈ V | 
| 6 |  | fvex 6918 | . . . . . . 7
⊢
(2nd ‘𝑥) ∈ V | 
| 7 | 5, 6 | opelrn 5953 | . . . . . 6
⊢
(〈(1st ‘𝑥), (2nd ‘𝑥)〉 ∈ 𝐴 → (2nd ‘𝑥) ∈ ran 𝐴) | 
| 8 | 4, 7 | syl 17 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ ran 𝐴) | 
| 9 |  | pwuninel 8301 | . . . . . 6
⊢  ¬
𝒫 ∪ ran 𝐴 ∈ ran 𝐴 | 
| 10 |  | xp2nd 8048 | . . . . . . . . 9
⊢ (𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴}) | 
| 11 | 10 | ad2antll 729 | . . . . . . . 8
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴}) | 
| 12 |  | elsni 4642 | . . . . . . . 8
⊢
((2nd ‘𝑥) ∈ {𝒫 ∪ ran 𝐴} → (2nd ‘𝑥) = 𝒫 ∪ ran 𝐴) | 
| 13 | 11, 12 | syl 17 | . . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → (2nd ‘𝑥) = 𝒫 ∪ ran 𝐴) | 
| 14 | 13 | eleq1d 2825 | . . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ((2nd ‘𝑥) ∈ ran 𝐴 ↔ 𝒫 ∪ ran 𝐴 ∈ ran 𝐴)) | 
| 15 | 9, 14 | mtbiri 327 | . . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ¬ (2nd
‘𝑥) ∈ ran 𝐴) | 
| 16 | 8, 15 | pm2.65da 816 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) | 
| 17 |  | elin 3966 | . . . 4
⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) | 
| 18 | 16, 17 | sylnibr 329 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴}))) | 
| 19 | 18 | eq0rdv 4406 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅) | 
| 20 |  | simpr 484 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | 
| 21 |  | rnexg 7925 | . . . . 5
⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) | 
| 22 | 21 | adantr 480 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V) | 
| 23 |  | uniexg 7761 | . . . 4
⊢ (ran
𝐴 ∈ V → ∪ ran 𝐴 ∈ V) | 
| 24 |  | pwexg 5377 | . . . 4
⊢ (∪ ran 𝐴 ∈ V → 𝒫 ∪ ran 𝐴 ∈ V) | 
| 25 | 22, 23, 24 | 3syl 18 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 ∪ ran 𝐴 ∈ V) | 
| 26 |  | xpsneng 9097 | . . 3
⊢ ((𝐵 ∈ 𝑊 ∧ 𝒫 ∪ ran 𝐴 ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵) | 
| 27 | 20, 25, 26 | syl2anc 584 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵) | 
| 28 | 19, 27 | jca 511 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)) |