Step | Hyp | Ref
| Expression |
1 | | 1st2nd2 7725 |
. . . . . . . 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 771 |
. . . . . . 7
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 𝑥 ∈ 𝐴) |
4 | 2, 3 | eqeltrrd 2852 |
. . . . . 6
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∈
𝐴) |
5 | | fvex 6664 |
. . . . . . 7
⊢
(1st ‘𝑥) ∈ V |
6 | | fvex 6664 |
. . . . . . 7
⊢
(2nd ‘𝑥) ∈ V |
7 | 5, 6 | opelrn 5777 |
. . . . . 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 7944 |
. . . . . 6
⊢ ¬
𝒫 ∪ ran 𝐴 ∈ ran 𝐴 |
10 | | xp2nd 7719 |
. . . . . . . . 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 4532 |
. . . . . . . 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 2835 |
. . . . . 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 331 |
. . . . 5
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) → ¬ (2nd
‘𝑥) ∈ ran 𝐴) |
16 | 8, 15 | pm2.65da 817 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) |
17 | | elin 3870 |
. . . 4
⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × {𝒫 ∪ ran 𝐴}))) |
18 | 16, 17 | sylnibr 333 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ 𝑥 ∈ (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴}))) |
19 | 18 | eq0rdv 4294 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅) |
20 | | simpr 489 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) |
21 | | rnexg 7607 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) |
22 | 21 | adantr 485 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V) |
23 | | uniexg 7457 |
. . . 4
⊢ (ran
𝐴 ∈ V → ∪ ran 𝐴 ∈ V) |
24 | | pwexg 5240 |
. . . 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 8615 |
. . 3
⊢ ((𝐵 ∈ 𝑊 ∧ 𝒫 ∪ ran 𝐴 ∈ V) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵) |
27 | 20, 25, 26 | syl2anc 588 |
. 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵) |
28 | 19, 27 | jca 516 |
1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵)) |