Step | Hyp | Ref
| Expression |
1 | | ovexd 7193 |
. 2
⊢ (𝜑 → (𝐴 ↑m {𝐵}) ∈ V) |
2 | | mapsnend.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
3 | 2 | elexd 3516 |
. 2
⊢ (𝜑 → 𝐴 ∈ V) |
4 | | fvexd 6687 |
. . 3
⊢ (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V) |
5 | 4 | a1i 11 |
. 2
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V)) |
6 | | snex 5334 |
. . 3
⊢
{〈𝐵, 𝑤〉} ∈
V |
7 | 6 | 2a1i 12 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V)) |
8 | | mapsnend.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
9 | 2, 8 | mapsnd 8452 |
. . . . . 6
⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}}) |
10 | 9 | abeq2d 2949 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉})) |
11 | 10 | anbi1d 631 |
. . . 4
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
12 | | r19.41v 3349 |
. . . . . 6
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
13 | 12 | bicomi 226 |
. . . . 5
⊢
((∃𝑦 ∈
𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
14 | 13 | a1i 11 |
. . . 4
⊢ (𝜑 → ((∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
15 | | df-rex 3146 |
. . . . 5
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
16 | 15 | a1i 11 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
17 | 11, 14, 16 | 3bitrd 307 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
18 | | fveq1 6671 |
. . . . . . . . . 10
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
19 | | vex 3499 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
20 | | fvsng 6944 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
21 | 8, 19, 20 | sylancl 588 |
. . . . . . . . . 10
⊢ (𝜑 → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
22 | 18, 21 | sylan9eqr 2880 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑧‘𝐵) = 𝑦) |
23 | 22 | eqeq2d 2834 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦)) |
24 | | equcom 2025 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) |
25 | 23, 24 | syl6bb 289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤)) |
26 | 25 | pm5.32da 581 |
. . . . . 6
⊢ (𝜑 → ((𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
27 | 26 | anbi2d 630 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
28 | | anass 471 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
29 | 28 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
30 | | ancom 463 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
31 | 30 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
32 | 27, 29, 31 | 3bitr2d 309 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
33 | 32 | exbidv 1922 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
34 | | eleq1w 2897 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) |
35 | | opeq2 4806 |
. . . . . . . 8
⊢ (𝑦 = 𝑤 → 〈𝐵, 𝑦〉 = 〈𝐵, 𝑤〉) |
36 | 35 | sneqd 4581 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → {〈𝐵, 𝑦〉} = {〈𝐵, 𝑤〉}) |
37 | 36 | eqeq2d 2834 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑧 = {〈𝐵, 𝑦〉} ↔ 𝑧 = {〈𝐵, 𝑤〉})) |
38 | 34, 37 | anbi12d 632 |
. . . . 5
⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
39 | 38 | equsexvw 2011 |
. . . 4
⊢
(∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
40 | 39 | a1i 11 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
41 | 17, 33, 40 | 3bitrd 307 |
. 2
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
42 | 1, 3, 5, 7, 41 | en2d 8547 |
1
⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴) |