Step | Hyp | Ref
| Expression |
1 | | ovex 6577 |
. . 3
⊢ (𝐴 ↑𝑚
{𝐵}) ∈
V |
2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ∈ V) |
3 | | mapsnend.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
4 | 3 | elexd 3187 |
. 2
⊢ (𝜑 → 𝐴 ∈ V) |
5 | | fvex 6113 |
. . . 4
⊢ (𝑧‘𝐵) ∈ V |
6 | 5 | a1i 11 |
. . 3
⊢ (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V) |
7 | 6 | a1i 11 |
. 2
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) → (𝑧‘𝐵) ∈ V)) |
8 | | snex 4835 |
. . . 4
⊢
{〈𝐵, 𝑤〉} ∈
V |
9 | 8 | a1i 11 |
. . 3
⊢ (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V) |
10 | 9 | a1i 11 |
. 2
⊢ (𝜑 → (𝑤 ∈ 𝐴 → {〈𝐵, 𝑤〉} ∈ V)) |
11 | | mapsnend.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
12 | 3, 11 | mapsnd 38383 |
. . . . . 6
⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉}}) |
13 | 12 | abeq2d 2721 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ↔ ∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉})) |
14 | 13 | anbi1d 737 |
. . . 4
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
15 | | r19.41v 3070 |
. . . . . 6
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
16 | 15 | bicomi 213 |
. . . . 5
⊢
((∃𝑦 ∈
𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) |
17 | 16 | a1i 11 |
. . . 4
⊢ (𝜑 → ((∃𝑦 ∈ 𝐴 𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
18 | | df-rex 2902 |
. . . . 5
⊢
(∃𝑦 ∈
𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)))) |
19 | 18 | a1i 11 |
. . . 4
⊢ (𝜑 → (∃𝑦 ∈ 𝐴 (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
20 | 14, 17, 19 | 3bitrd 293 |
. . 3
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))))) |
21 | | fveq1 6102 |
. . . . . . . . . . . 12
⊢ (𝑧 = {〈𝐵, 𝑦〉} → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
22 | 21 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑧‘𝐵) = ({〈𝐵, 𝑦〉}‘𝐵)) |
23 | | vex 3176 |
. . . . . . . . . . . . . 14
⊢ 𝑦 ∈ V |
24 | 23 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑦 ∈ V) |
25 | | fvsng 6352 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
26 | 11, 24, 25 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (𝜑 → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
27 | 26 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → ({〈𝐵, 𝑦〉}‘𝐵) = 𝑦) |
28 | 22, 27 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑧‘𝐵) = 𝑦) |
29 | 28 | eqeq2d 2620 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑤 = 𝑦)) |
30 | | equcom 1932 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤) |
31 | 30 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)) |
32 | 29, 31 | bitrd 267 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = {〈𝐵, 𝑦〉}) → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤)) |
33 | 32 | ex 449 |
. . . . . . 7
⊢ (𝜑 → (𝑧 = {〈𝐵, 𝑦〉} → (𝑤 = (𝑧‘𝐵) ↔ 𝑦 = 𝑤))) |
34 | 33 | pm5.32d 669 |
. . . . . 6
⊢ (𝜑 → ((𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
35 | 34 | anbi2d 736 |
. . . . 5
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
36 | | anass 679 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤))) |
37 | 36 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑦 = 𝑤)))) |
38 | | ancom 465 |
. . . . . 6
⊢ (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}))) |
39 | 38 | a1i 11 |
. . . . 5
⊢ (𝜑 → (((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ∧ 𝑦 = 𝑤) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
40 | 35, 37, 39 | 3bitr2d 295 |
. . . 4
⊢ (𝜑 → ((𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ (𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
41 | 40 | exbidv 1837 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑧 = {〈𝐵, 𝑦〉} ∧ 𝑤 = (𝑧‘𝐵))) ↔ ∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})))) |
42 | | vex 3176 |
. . . . 5
⊢ 𝑤 ∈ V |
43 | | eleq1 2676 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴)) |
44 | | opeq2 4341 |
. . . . . . . 8
⊢ (𝑦 = 𝑤 → 〈𝐵, 𝑦〉 = 〈𝐵, 𝑤〉) |
45 | 44 | sneqd 4137 |
. . . . . . 7
⊢ (𝑦 = 𝑤 → {〈𝐵, 𝑦〉} = {〈𝐵, 𝑤〉}) |
46 | 45 | eqeq2d 2620 |
. . . . . 6
⊢ (𝑦 = 𝑤 → (𝑧 = {〈𝐵, 𝑦〉} ↔ 𝑧 = {〈𝐵, 𝑤〉})) |
47 | 43, 46 | anbi12d 743 |
. . . . 5
⊢ (𝑦 = 𝑤 → ((𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉}) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
48 | 42, 47 | ceqsexv 3215 |
. . . 4
⊢
(∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉})) |
49 | 48 | a1i 11 |
. . 3
⊢ (𝜑 → (∃𝑦(𝑦 = 𝑤 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑦〉})) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
50 | 20, 41, 49 | 3bitrd 293 |
. 2
⊢ (𝜑 → ((𝑧 ∈ (𝐴 ↑𝑚 {𝐵}) ∧ 𝑤 = (𝑧‘𝐵)) ↔ (𝑤 ∈ 𝐴 ∧ 𝑧 = {〈𝐵, 𝑤〉}))) |
51 | 2, 4, 7, 10, 50 | en2d 7877 |
1
⊢ (𝜑 → (𝐴 ↑𝑚 {𝐵}) ≈ 𝐴) |