Step | Hyp | Ref
| Expression |
1 | | ovex 6678 |
. 2
⊢ (𝐴 ↑𝑚
{𝐵}) ∈
V |
2 | | mapsnen.1 |
. 2
⊢ 𝐴 ∈ V |
3 | | fvexd 6203 |
. 2
⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V) |
4 | | snex 4908 |
. . 3
⊢
{〈𝐵, 𝑤〉} ∈
V |
5 | 4 | a1i 11 |
. 2
⊢ (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V) |
6 | | mapsnen.2 |
. . . . . . 7
⊢ 𝐵 ∈ V |
7 | 2, 6 | mapsn 7899 |
. . . . . 6
⊢ (𝐴 ↑𝑚
{𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}} |
8 | 7 | abeq2i 2735 |
. . . . 5
⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}) |
9 | 8 | anbi1i 731 |
. . . 4
⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
10 | | r19.41v 3089 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
11 | | df-rex 2918 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
12 | 9, 10, 11 | 3bitr2i 288 |
. . 3
⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
13 | | fveq1 6190 |
. . . . . . . . . 10
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
14 | | vex 3203 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
15 | 6, 14 | fvsn 6446 |
. . . . . . . . . 10
⊢
({〈𝐵, 𝑦〉}‘𝐵) = 𝑦 |
16 | 13, 15 | syl6eq 2672 |
. . . . . . . . 9
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = 𝑦) |
17 | 16 | eqeq2d 2632 |
. . . . . . . 8
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦)) |
18 | | equcom 1945 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) |
19 | 17, 18 | syl6bb 276 |
. . . . . . 7
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤)) |
20 | 19 | pm5.32i 669 |
. . . . . 6
⊢ ((𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)) |
21 | 20 | anbi2i 730 |
. . . . 5
⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
22 | | anass 681 |
. . . . 5
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
23 | | ancom 466 |
. . . . 5
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
24 | 21, 22, 23 | 3bitr2i 288 |
. . . 4
⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
25 | 24 | exbii 1774 |
. . 3
⊢
(∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
26 | | vex 3203 |
. . . 4
⊢ 𝑤 ∈ V |
27 | | eleq1 2689 |
. . . . 5
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) |
28 | | opeq2 4403 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → 〈𝐵, 𝑦〉 = 〈𝐵, 𝑤〉) |
29 | 28 | sneqd 4189 |
. . . . . 6
⊢ (𝑦 = 𝑤 → {〈𝐵, 𝑦〉} = {〈𝐵, 𝑤〉}) |
30 | 29 | eqeq2d 2632 |
. . . . 5
⊢ (𝑦 = 𝑤 → (𝑧 = {〈𝐵, 𝑦〉} ↔ 𝑧 = {〈𝐵, 𝑤〉})) |
31 | 27, 30 | anbi12d 747 |
. . . 4
⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
32 | 26, 31 | ceqsexv 3242 |
. . 3
⊢
(∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
33 | 12, 25, 32 | 3bitri 286 |
. 2
⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
34 | 1, 2, 3, 5, 33 | en2i 7993 |
1
⊢ (𝐴 ↑𝑚
{𝐵}) ≈ 𝐴 |