Step | Hyp | Ref
| Expression |
1 | | relopab 5484 |
. . 3
⊢ Rel
{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} |
2 | | ancom 454 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) |
3 | | eqcom 2832 |
. . . . . 6
⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)) |
4 | 2, 3 | anbi12i 620 |
. . . . 5
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑥))) |
5 | | simpl 476 |
. . . . . . . 8
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → 𝑢 = 𝑥) |
6 | 5 | fveq2d 6441 |
. . . . . . 7
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝐹‘𝑢) = (𝐹‘𝑥)) |
7 | | simpr 479 |
. . . . . . . 8
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → 𝑣 = 𝑦) |
8 | 7 | fveq2d 6441 |
. . . . . . 7
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝐹‘𝑣) = (𝐹‘𝑦)) |
9 | 6, 8 | eqeq12d 2840 |
. . . . . 6
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) |
10 | | eqid 2825 |
. . . . . 6
⊢
{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} |
11 | 9, 10 | brab2a 5433 |
. . . . 5
⊢ (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) |
12 | | simpl 476 |
. . . . . . . 8
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → 𝑢 = 𝑦) |
13 | 12 | fveq2d 6441 |
. . . . . . 7
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝐹‘𝑢) = (𝐹‘𝑦)) |
14 | | simpr 479 |
. . . . . . . 8
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → 𝑣 = 𝑥) |
15 | 14 | fveq2d 6441 |
. . . . . . 7
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝐹‘𝑣) = (𝐹‘𝑥)) |
16 | 13, 15 | eqeq12d 2840 |
. . . . . 6
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))) |
17 | 16, 10 | brab2a 5433 |
. . . . 5
⊢ (𝑦{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥 ↔ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑥))) |
18 | 4, 11, 17 | 3bitr4i 295 |
. . . 4
⊢ (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ 𝑦{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥) |
19 | 18 | biimpi 208 |
. . 3
⊢ (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 → 𝑦{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥) |
20 | | simplll 791 |
. . . . . 6
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → 𝑥 ∈ 𝐴) |
21 | | simprlr 798 |
. . . . . 6
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → 𝑧 ∈ 𝐴) |
22 | 20, 21 | jca 507 |
. . . . 5
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) |
23 | | simplr 785 |
. . . . . 6
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑥) = (𝐹‘𝑦)) |
24 | | simprr 789 |
. . . . . 6
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑦) = (𝐹‘𝑧)) |
25 | 23, 24 | eqtrd 2861 |
. . . . 5
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → (𝐹‘𝑥) = (𝐹‘𝑧)) |
26 | 22, 25 | jca 507 |
. . . 4
⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑧))) |
27 | | simpl 476 |
. . . . . . . 8
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → 𝑢 = 𝑦) |
28 | 27 | fveq2d 6441 |
. . . . . . 7
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → (𝐹‘𝑢) = (𝐹‘𝑦)) |
29 | | simpr 479 |
. . . . . . . 8
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧) |
30 | 29 | fveq2d 6441 |
. . . . . . 7
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → (𝐹‘𝑣) = (𝐹‘𝑧)) |
31 | 28, 30 | eqeq12d 2840 |
. . . . . 6
⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑧) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑦) = (𝐹‘𝑧))) |
32 | 31, 10 | brab2a 5433 |
. . . . 5
⊢ (𝑦{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧 ↔ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧))) |
33 | 11, 32 | anbi12i 620 |
. . . 4
⊢ ((𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ∧ 𝑦{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) ∧ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑦) = (𝐹‘𝑧)))) |
34 | | simpl 476 |
. . . . . . 7
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → 𝑢 = 𝑥) |
35 | 34 | fveq2d 6441 |
. . . . . 6
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → (𝐹‘𝑢) = (𝐹‘𝑥)) |
36 | | simpr 479 |
. . . . . . 7
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧) |
37 | 36 | fveq2d 6441 |
. . . . . 6
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → (𝐹‘𝑣) = (𝐹‘𝑧)) |
38 | 35, 37 | eqeq12d 2840 |
. . . . 5
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑧) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑧))) |
39 | 38, 10 | brab2a 5433 |
. . . 4
⊢ (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑧))) |
40 | 26, 33, 39 | 3imtr4i 284 |
. . 3
⊢ ((𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ∧ 𝑦{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧) → 𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑧) |
41 | | pm4.24 559 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) |
42 | | eqid 2825 |
. . . . . 6
⊢ (𝐹‘𝑥) = (𝐹‘𝑥) |
43 | 42 | biantru 525 |
. . . . 5
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑥))) |
44 | 41, 43 | bitri 267 |
. . . 4
⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑥))) |
45 | | simpl 476 |
. . . . . . 7
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → 𝑢 = 𝑥) |
46 | 45 | fveq2d 6441 |
. . . . . 6
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → (𝐹‘𝑢) = (𝐹‘𝑥)) |
47 | | simpr 479 |
. . . . . . 7
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → 𝑣 = 𝑥) |
48 | 47 | fveq2d 6441 |
. . . . . 6
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → (𝐹‘𝑣) = (𝐹‘𝑥)) |
49 | 46, 48 | eqeq12d 2840 |
. . . . 5
⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑥) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑥) = (𝐹‘𝑥))) |
50 | 49, 10 | brab2a 5433 |
. . . 4
⊢ (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑥))) |
51 | 44, 50 | bitr4i 270 |
. . 3
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑥) |
52 | 1, 19, 40, 51 | iseri 8041 |
. 2
⊢
{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 |
53 | 11 | baib 531 |
. . 3
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) |
54 | 53 | rgen2 3184 |
. 2
⊢
∀𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐴 (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)) |
55 | | id 22 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉) |
56 | | simprll 797 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))) → 𝑢 ∈ 𝐴) |
57 | | simprlr 798 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))) → 𝑣 ∈ 𝐴) |
58 | 55, 55, 56, 57 | opabex2 7494 |
. . 3
⊢ (𝐴 ∈ 𝑉 → {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∈ V) |
59 | | ereq1 8021 |
. . . . 5
⊢ (𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} → (𝑟 Er 𝐴 ↔ {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴)) |
60 | | simpl 476 |
. . . . . . . 8
⊢ ((𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}) |
61 | 60 | breqd 4886 |
. . . . . . 7
⊢ ((𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑟𝑦 ↔ 𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦)) |
62 | 61 | bibi1d 335 |
. . . . . 6
⊢ ((𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))) |
63 | 62 | 2ralbidva 3197 |
. . . . 5
⊢ (𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))) |
64 | 59, 63 | anbi12d 624 |
. . . 4
⊢ (𝑟 = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} → ((𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) ↔ ({〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))) |
65 | 64 | spcegv 3511 |
. . 3
⊢
({〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} ∈ V → (({〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))) |
66 | 58, 65 | syl 17 |
. 2
⊢ (𝐴 ∈ 𝑉 → (({〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))} Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥{〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝐹‘𝑢) = (𝐹‘𝑣))}𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))) → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦))))) |
67 | 52, 54, 66 | mp2ani 689 |
1
⊢ (𝐴 ∈ 𝑉 → ∃𝑟(𝑟 Er 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑟𝑦 ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))) |