Step | Hyp | Ref
| Expression |
1 | | vex 3444 |
. . . . 5
⊢ 𝑥 ∈ V |
2 | | vex 3444 |
. . . . 5
⊢ 𝑦 ∈ V |
3 | 1, 2 | brcnv 5717 |
. . . 4
⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦(1st ↾ I )𝑥) |
4 | 1 | brresi 5827 |
. . . . 5
⊢ (𝑦(1st ↾ I )𝑥 ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥)) |
5 | | 19.42v 1954 |
. . . . . . 7
⊢
(∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉) ↔ ((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = 〈𝑧, 𝑧〉)) |
6 | | vex 3444 |
. . . . . . . . . . 11
⊢ 𝑧 ∈ V |
7 | 6, 6 | op1std 7681 |
. . . . . . . . . 10
⊢ (𝑦 = 〈𝑧, 𝑧〉 → (1st ‘𝑦) = 𝑧) |
8 | 7 | eqeq1d 2800 |
. . . . . . . . 9
⊢ (𝑦 = 〈𝑧, 𝑧〉 → ((1st ‘𝑦) = 𝑥 ↔ 𝑧 = 𝑥)) |
9 | 8 | pm5.32ri 579 |
. . . . . . . 8
⊢
(((1st ‘𝑦) = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉) ↔ (𝑧 = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉)) |
10 | 9 | exbii 1849 |
. . . . . . 7
⊢
(∃𝑧((1st ‘𝑦) = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉)) |
11 | | fo1st 7691 |
. . . . . . . . . 10
⊢
1st :V–onto→V |
12 | | fofn 6567 |
. . . . . . . . . 10
⊢
(1st :V–onto→V → 1st Fn V) |
13 | 11, 12 | ax-mp 5 |
. . . . . . . . 9
⊢
1st Fn V |
14 | | fnbrfvb 6693 |
. . . . . . . . 9
⊢
((1st Fn V ∧ 𝑦 ∈ V) → ((1st
‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥)) |
15 | 13, 2, 14 | mp2an 691 |
. . . . . . . 8
⊢
((1st ‘𝑦) = 𝑥 ↔ 𝑦1st 𝑥) |
16 | | dfid2 5428 |
. . . . . . . . . 10
⊢ I =
{〈𝑧, 𝑧〉 ∣ 𝑧 = 𝑧} |
17 | 16 | eleq2i 2881 |
. . . . . . . . 9
⊢ (𝑦 ∈ I ↔ 𝑦 ∈ {〈𝑧, 𝑧〉 ∣ 𝑧 = 𝑧}) |
18 | | nfe1 2151 |
. . . . . . . . . . 11
⊢
Ⅎ𝑧∃𝑧(𝑦 = 〈𝑧, 𝑧〉 ∧ 𝑧 = 𝑧) |
19 | 18 | 19.9 2203 |
. . . . . . . . . 10
⊢
(∃𝑧∃𝑧(𝑦 = 〈𝑧, 𝑧〉 ∧ 𝑧 = 𝑧) ↔ ∃𝑧(𝑦 = 〈𝑧, 𝑧〉 ∧ 𝑧 = 𝑧)) |
20 | | elopab 5379 |
. . . . . . . . . 10
⊢ (𝑦 ∈ {〈𝑧, 𝑧〉 ∣ 𝑧 = 𝑧} ↔ ∃𝑧∃𝑧(𝑦 = 〈𝑧, 𝑧〉 ∧ 𝑧 = 𝑧)) |
21 | | equid 2019 |
. . . . . . . . . . . 12
⊢ 𝑧 = 𝑧 |
22 | 21 | biantru 533 |
. . . . . . . . . . 11
⊢ (𝑦 = 〈𝑧, 𝑧〉 ↔ (𝑦 = 〈𝑧, 𝑧〉 ∧ 𝑧 = 𝑧)) |
23 | 22 | exbii 1849 |
. . . . . . . . . 10
⊢
(∃𝑧 𝑦 = 〈𝑧, 𝑧〉 ↔ ∃𝑧(𝑦 = 〈𝑧, 𝑧〉 ∧ 𝑧 = 𝑧)) |
24 | 19, 20, 23 | 3bitr4i 306 |
. . . . . . . . 9
⊢ (𝑦 ∈ {〈𝑧, 𝑧〉 ∣ 𝑧 = 𝑧} ↔ ∃𝑧 𝑦 = 〈𝑧, 𝑧〉) |
25 | 17, 24 | bitr2i 279 |
. . . . . . . 8
⊢
(∃𝑧 𝑦 = 〈𝑧, 𝑧〉 ↔ 𝑦 ∈ I ) |
26 | 15, 25 | anbi12ci 630 |
. . . . . . 7
⊢
(((1st ‘𝑦) = 𝑥 ∧ ∃𝑧 𝑦 = 〈𝑧, 𝑧〉) ↔ (𝑦 ∈ I ∧ 𝑦1st 𝑥)) |
27 | 5, 10, 26 | 3bitr3ri 305 |
. . . . . 6
⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉)) |
28 | | id 22 |
. . . . . . . . 9
⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥) |
29 | 28, 28 | opeq12d 4773 |
. . . . . . . 8
⊢ (𝑧 = 𝑥 → 〈𝑧, 𝑧〉 = 〈𝑥, 𝑥〉) |
30 | 29 | eqeq2d 2809 |
. . . . . . 7
⊢ (𝑧 = 𝑥 → (𝑦 = 〈𝑧, 𝑧〉 ↔ 𝑦 = 〈𝑥, 𝑥〉)) |
31 | 30 | equsexvw 2011 |
. . . . . 6
⊢
(∃𝑧(𝑧 = 𝑥 ∧ 𝑦 = 〈𝑧, 𝑧〉) ↔ 𝑦 = 〈𝑥, 𝑥〉) |
32 | 27, 31 | bitri 278 |
. . . . 5
⊢ ((𝑦 ∈ I ∧ 𝑦1st 𝑥) ↔ 𝑦 = 〈𝑥, 𝑥〉) |
33 | 4, 32 | bitri 278 |
. . . 4
⊢ (𝑦(1st ↾ I )𝑥 ↔ 𝑦 = 〈𝑥, 𝑥〉) |
34 | 3, 33 | bitri 278 |
. . 3
⊢ (𝑥◡(1st ↾ I )𝑦 ↔ 𝑦 = 〈𝑥, 𝑥〉) |
35 | 34 | opabbii 5097 |
. 2
⊢
{〈𝑥, 𝑦〉 ∣ 𝑥◡(1st ↾ I )𝑦} = {〈𝑥, 𝑦〉 ∣ 𝑦 = 〈𝑥, 𝑥〉} |
36 | | relcnv 5934 |
. . 3
⊢ Rel ◡(1st ↾ I ) |
37 | | dfrel4v 6014 |
. . 3
⊢ (Rel
◡(1st ↾ I ) ↔
◡(1st ↾ I ) =
{〈𝑥, 𝑦〉 ∣ 𝑥◡(1st ↾ I )𝑦}) |
38 | 36, 37 | mpbi 233 |
. 2
⊢ ◡(1st ↾ I ) = {〈𝑥, 𝑦〉 ∣ 𝑥◡(1st ↾ I )𝑦} |
39 | | mptv 5135 |
. 2
⊢ (𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) = {〈𝑥, 𝑦〉 ∣ 𝑦 = 〈𝑥, 𝑥〉} |
40 | 35, 38, 39 | 3eqtr4i 2831 |
1
⊢ ◡(1st ↾ I ) = (𝑥 ∈ V ↦ 〈𝑥, 𝑥〉) |