Step | Hyp | Ref
| Expression |
1 | | ima0 5910 |
. . . 4
⊢
(2nd “ ∅) = ∅ |
2 | | xpeq2 5538 |
. . . . . 6
⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) |
3 | | xp0 5980 |
. . . . . 6
⊢ (𝐴 × ∅) =
∅ |
4 | 2, 3 | eqtrdi 2810 |
. . . . 5
⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
5 | 4 | imaeq2d 5894 |
. . . 4
⊢ (𝐵 = ∅ →
(2nd “ (𝐴
× 𝐵)) =
(2nd “ ∅)) |
6 | | id 22 |
. . . 4
⊢ (𝐵 = ∅ → 𝐵 = ∅) |
7 | 1, 5, 6 | 3eqtr4a 2820 |
. . 3
⊢ (𝐵 = ∅ →
(2nd “ (𝐴
× 𝐵)) = 𝐵) |
8 | 7 | adantl 486 |
. 2
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 = ∅) →
(2nd “ (𝐴
× 𝐵)) = 𝐵) |
9 | | xpnz 5981 |
. . . . 5
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅) |
10 | | fo2nd 7707 |
. . . . . . 7
⊢
2nd :V–onto→V |
11 | | fofn 6571 |
. . . . . . 7
⊢
(2nd :V–onto→V → 2nd Fn V) |
12 | 10, 11 | mp1i 13 |
. . . . . 6
⊢ ((𝐴 × 𝐵) ≠ ∅ → 2nd Fn
V) |
13 | | ssv 3912 |
. . . . . . 7
⊢ (𝐴 × 𝐵) ⊆ V |
14 | 13 | a1i 11 |
. . . . . 6
⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) ⊆ V) |
15 | 12, 14 | fvelimabd 6719 |
. . . . 5
⊢ ((𝐴 × 𝐵) ≠ ∅ → (𝑦 ∈ (2nd “ (𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)) |
16 | 9, 15 | sylbi 220 |
. . . 4
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “
(𝐴 × 𝐵)) ↔ ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦)) |
17 | | simpr 489 |
. . . . . . 7
⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) = 𝑦) |
18 | | xp2nd 7719 |
. . . . . . . 8
⊢ (𝑝 ∈ (𝐴 × 𝐵) → (2nd ‘𝑝) ∈ 𝐵) |
19 | 18 | ad2antlr 727 |
. . . . . . 7
⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → (2nd ‘𝑝) ∈ 𝐵) |
20 | 17, 19 | eqeltrrd 2852 |
. . . . . 6
⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑝 ∈ (𝐴 × 𝐵)) ∧ (2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵) |
21 | 20 | r19.29an 3210 |
. . . . 5
⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧
∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) → 𝑦 ∈ 𝐵) |
22 | | n0 4239 |
. . . . . . . 8
⊢ (𝐴 ≠ ∅ ↔
∃𝑥 𝑥 ∈ 𝐴) |
23 | 22 | biimpi 219 |
. . . . . . 7
⊢ (𝐴 ≠ ∅ →
∃𝑥 𝑥 ∈ 𝐴) |
24 | 23 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴) |
25 | | opelxpi 5554 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
26 | 25 | ancoms 463 |
. . . . . . . 8
⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
27 | 26 | adantll 714 |
. . . . . . 7
⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵)) |
28 | | fveqeq2 6660 |
. . . . . . . 8
⊢ (𝑝 = 〈𝑥, 𝑦〉 → ((2nd ‘𝑝) = 𝑦 ↔ (2nd ‘〈𝑥, 𝑦〉) = 𝑦)) |
29 | 28 | adantl 486 |
. . . . . . 7
⊢
(((((𝐴 ≠ ∅
∧ 𝐵 ≠ ∅) ∧
𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑝 = 〈𝑥, 𝑦〉) → ((2nd ‘𝑝) = 𝑦 ↔ (2nd ‘〈𝑥, 𝑦〉) = 𝑦)) |
30 | | vex 3411 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
31 | | vex 3411 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
32 | 30, 31 | op2nd 7695 |
. . . . . . . 8
⊢
(2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
33 | 32 | a1i 11 |
. . . . . . 7
⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (2nd ‘〈𝑥, 𝑦〉) = 𝑦) |
34 | 27, 29, 33 | rspcedvd 3542 |
. . . . . 6
⊢ ((((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) |
35 | 24, 34 | exlimddv 1937 |
. . . . 5
⊢ (((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ∧ 𝑦 ∈ 𝐵) → ∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦) |
36 | 21, 35 | impbida 801 |
. . . 4
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) →
(∃𝑝 ∈ (𝐴 × 𝐵)(2nd ‘𝑝) = 𝑦 ↔ 𝑦 ∈ 𝐵)) |
37 | 16, 36 | bitrd 282 |
. . 3
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → (𝑦 ∈ (2nd “
(𝐴 × 𝐵)) ↔ 𝑦 ∈ 𝐵)) |
38 | 37 | eqrdv 2757 |
. 2
⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) →
(2nd “ (𝐴
× 𝐵)) = 𝐵) |
39 | 8, 38 | pm2.61dane 3036 |
1
⊢ (𝐴 ≠ ∅ →
(2nd “ (𝐴
× 𝐵)) = 𝐵) |