Step | Hyp | Ref
| Expression |
1 | | ovexd 6680 |
. 2
⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ∈ V) |
2 | | mapsnend.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
3 | 2 | elexd 3214 |
. 2
⊢ (𝜑 → 𝐴 ∈ V) |
4 | | fvexd 6203 |
. . 3
⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V) |
5 | 4 | a1i 11 |
. 2
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V)) |
6 | | snex 4908 |
. . . 4
⊢
{〈𝐵, 𝑤〉} ∈
V |
7 | 6 | a1i 11 |
. . 3
⊢ (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V) |
8 | 7 | a1i 11 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V)) |
9 | | mapsnend.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
10 | 2, 9 | mapsnd 39388 |
. . . . . 6
⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}}) |
11 | 10 | abeq2d 2734 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉})) |
12 | 11 | anbi1d 741 |
. . . 4
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
13 | | r19.41v 3089 |
. . . . . 6
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
14 | 13 | bicomi 214 |
. . . . 5
⊢
((∃𝑦 ∈
𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
15 | 14 | a1i 11 |
. . . 4
⊢ (𝜑 → ((∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
16 | | df-rex 2918 |
. . . . 5
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
17 | 16 | a1i 11 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
18 | 12, 15, 17 | 3bitrd 294 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
19 | | fveq1 6190 |
. . . . . . . . . . . 12
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
20 | 19 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
21 | | vex 3203 |
. . . . . . . . . . . . . 14
⊢ 𝑦 ∈ V |
22 | 21 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑦 ∈ V) |
23 | | fvsng 6447 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
24 | 9, 22, 23 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (𝜑 → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
25 | 24 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
26 | 20, 25 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑧‘𝐵) = 𝑦) |
27 | 26 | eqeq2d 2632 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦)) |
28 | | equcom 1945 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) |
29 | 28 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)) |
30 | 27, 29 | bitrd 268 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤)) |
31 | 30 | ex 450 |
. . . . . . 7
⊢ (𝜑 → (𝑧 = {〈𝐵, 𝑦〉} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))) |
32 | 31 | pm5.32d 671 |
. . . . . 6
⊢ (𝜑 → ((𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
33 | 32 | anbi2d 740 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
34 | | anass 681 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
35 | 34 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
36 | | ancom 466 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
37 | 36 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
38 | 33, 35, 37 | 3bitr2d 296 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
39 | 38 | exbidv 1850 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
40 | | vex 3203 |
. . . . 5
⊢ 𝑤 ∈ V |
41 | | eleq1 2689 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) |
42 | | opeq2 4403 |
. . . . . . . 8
⊢ (𝑦 = 𝑤 → 〈𝐵, 𝑦〉 = 〈𝐵, 𝑤〉) |
43 | 42 | sneqd 4189 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → {〈𝐵, 𝑦〉} = {〈𝐵, 𝑤〉}) |
44 | 43 | eqeq2d 2632 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑧 = {〈𝐵, 𝑦〉} ↔ 𝑧 = {〈𝐵, 𝑤〉})) |
45 | 41, 44 | anbi12d 747 |
. . . . 5
⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
46 | 40, 45 | ceqsexv 3242 |
. . . 4
⊢
(∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
47 | 46 | a1i 11 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
48 | 18, 39, 47 | 3bitrd 294 |
. 2
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
49 | 1, 3, 5, 8, 48 | en2d 7991 |
1
⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴) |