Step | Hyp | Ref
| Expression |
1 | | fnmap 6633 |
. . 3
⊢
↑𝑚 Fn (V × V) |
2 | | mapsnen.1 |
. . 3
⊢ 𝐴 ∈ V |
3 | | mapsnen.2 |
. . . 4
⊢ 𝐵 ∈ V |
4 | 3 | snex 4171 |
. . 3
⊢ {𝐵} ∈ V |
5 | | fnovex 5886 |
. . 3
⊢ ((
↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ {𝐵} ∈ V) → (𝐴 ↑𝑚 {𝐵}) ∈ V) |
6 | 1, 2, 4, 5 | mp3an 1332 |
. 2
⊢ (𝐴 ↑𝑚
{𝐵}) ∈
V |
7 | | vex 2733 |
. . . 4
⊢ 𝑧 ∈ V |
8 | 7, 3 | fvex 5516 |
. . 3
⊢ (𝑧‘𝐵) ∈ V |
9 | 8 | a1i 9 |
. 2
⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V) |
10 | | vex 2733 |
. . . . 5
⊢ 𝑤 ∈ V |
11 | 3, 10 | opex 4214 |
. . . 4
⊢
〈𝐵, 𝑤〉 ∈ V |
12 | 11 | snex 4171 |
. . 3
⊢
{〈𝐵, 𝑤〉} ∈
V |
13 | 12 | a1i 9 |
. 2
⊢ (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V) |
14 | 2, 3 | mapsn 6668 |
. . . . . 6
⊢ (𝐴 ↑𝑚
{𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}} |
15 | 14 | abeq2i 2281 |
. . . . 5
⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}) |
16 | 15 | anbi1i 455 |
. . . 4
⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
17 | | r19.41v 2626 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
18 | | df-rex 2454 |
. . . 4
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
19 | 16, 17, 18 | 3bitr2i 207 |
. . 3
⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
20 | | fveq1 5495 |
. . . . . . . . . 10
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
21 | | vex 2733 |
. . . . . . . . . . 11
⊢ 𝑦 ∈ V |
22 | 3, 21 | fvsn 5691 |
. . . . . . . . . 10
⊢
({〈𝐵, 𝑦〉}‘𝐵) = 𝑦 |
23 | 20, 22 | eqtrdi 2219 |
. . . . . . . . 9
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = 𝑦) |
24 | 23 | eqeq2d 2182 |
. . . . . . . 8
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦)) |
25 | | equcom 1699 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) |
26 | 24, 25 | bitrdi 195 |
. . . . . . 7
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤)) |
27 | 26 | pm5.32i 451 |
. . . . . 6
⊢ ((𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)) |
28 | 27 | anbi2i 454 |
. . . . 5
⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
29 | | anass 399 |
. . . . 5
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
30 | | ancom 264 |
. . . . 5
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
31 | 28, 29, 30 | 3bitr2i 207 |
. . . 4
⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
32 | 31 | exbii 1598 |
. . 3
⊢
(∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
33 | | eleq1w 2231 |
. . . . 5
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) |
34 | | opeq2 3766 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → 〈𝐵, 𝑦〉 = 〈𝐵, 𝑤〉) |
35 | 34 | sneqd 3596 |
. . . . . 6
⊢ (𝑦 = 𝑤 → {〈𝐵, 𝑦〉} = {〈𝐵, 𝑤〉}) |
36 | 35 | eqeq2d 2182 |
. . . . 5
⊢ (𝑦 = 𝑤 → (𝑧 = {〈𝐵, 𝑦〉} ↔ 𝑧 = {〈𝐵, 𝑤〉})) |
37 | 33, 36 | anbi12d 470 |
. . . 4
⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
38 | 10, 37 | ceqsexv 2769 |
. . 3
⊢
(∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
39 | 19, 32, 38 | 3bitri 205 |
. 2
⊢ ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
40 | 6, 2, 9, 13, 39 | en2i 6748 |
1
⊢ (𝐴 ↑𝑚
{𝐵}) ≈ 𝐴 |