Step | Hyp | Ref
| Expression |
1 | | ovexd 7248 |
. 2
⊢ (𝜑 → (𝐴 ↑m {𝐵}) ∈ V) |
2 | | mapsnend.a |
. 2
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
3 | | fvexd 6732 |
. . 3
⊢ (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V) |
4 | 3 | a1i 11 |
. 2
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) → (𝑧‘𝐵) ∈ V)) |
5 | | snex 5324 |
. . 3
⊢
{〈𝐵, 𝑤〉} ∈
V |
6 | 5 | 2a1i 12 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V)) |
7 | | mapsnend.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
8 | 2, 7 | mapsnd 8567 |
. . . . . 6
⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}}) |
9 | 8 | abeq2d 2871 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑m {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉})) |
10 | 9 | anbi1d 633 |
. . . 4
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
11 | | r19.41v 3260 |
. . . . . 6
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
12 | 11 | bicomi 227 |
. . . . 5
⊢
((∃𝑦 ∈
𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
13 | 12 | a1i 11 |
. . . 4
⊢ (𝜑 → ((∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
14 | | df-rex 3067 |
. . . . 5
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
15 | 14 | a1i 11 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
16 | 10, 13, 15 | 3bitrd 308 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
17 | | fveq1 6716 |
. . . . . . . . . 10
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
18 | | vex 3412 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
19 | | fvsng 6995 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
20 | 7, 18, 19 | sylancl 589 |
. . . . . . . . . 10
⊢ (𝜑 → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
21 | 17, 20 | sylan9eqr 2800 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑧‘𝐵) = 𝑦) |
22 | 21 | eqeq2d 2748 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦)) |
23 | | equcom 2026 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) |
24 | 22, 23 | bitrdi 290 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤)) |
25 | 24 | pm5.32da 582 |
. . . . . 6
⊢ (𝜑 → ((𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
26 | 25 | anbi2d 632 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
27 | | anass 472 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
28 | 27 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
29 | | ancom 464 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
30 | 29 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
31 | 26, 28, 30 | 3bitr2d 310 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
32 | 31 | exbidv 1929 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
33 | | eleq1w 2820 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) |
34 | | opeq2 4785 |
. . . . . . . 8
⊢ (𝑦 = 𝑤 → 〈𝐵, 𝑦〉 = 〈𝐵, 𝑤〉) |
35 | 34 | sneqd 4553 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → {〈𝐵, 𝑦〉} = {〈𝐵, 𝑤〉}) |
36 | 35 | eqeq2d 2748 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑧 = {〈𝐵, 𝑦〉} ↔ 𝑧 = {〈𝐵, 𝑤〉})) |
37 | 33, 36 | anbi12d 634 |
. . . . 5
⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
38 | 37 | equsexvw 2013 |
. . . 4
⊢
(∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
39 | 38 | a1i 11 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
40 | 16, 32, 39 | 3bitrd 308 |
. 2
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑m {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
41 | 1, 2, 4, 6, 40 | en2d 8664 |
1
⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴) |