Step | Hyp | Ref
| Expression |
1 | | fveq2 6668 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅)) |
2 | | idlval.1 |
. . . . . . 7
⊢ 𝐺 = (1st ‘𝑅) |
3 | 1, 2 | eqtr4di 2791 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺) |
4 | 3 | rneqd 5775 |
. . . . 5
⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺) |
5 | | idlval.3 |
. . . . 5
⊢ 𝑋 = ran 𝐺 |
6 | 4, 5 | eqtr4di 2791 |
. . . 4
⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋) |
7 | 6 | pweqd 4504 |
. . 3
⊢ (𝑟 = 𝑅 → 𝒫 ran (1st
‘𝑟) = 𝒫 𝑋) |
8 | 3 | fveq2d 6672 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (GId‘(1st
‘𝑟)) =
(GId‘𝐺)) |
9 | | idlval.4 |
. . . . . 6
⊢ 𝑍 = (GId‘𝐺) |
10 | 8, 9 | eqtr4di 2791 |
. . . . 5
⊢ (𝑟 = 𝑅 → (GId‘(1st
‘𝑟)) = 𝑍) |
11 | 10 | eleq1d 2817 |
. . . 4
⊢ (𝑟 = 𝑅 → ((GId‘(1st
‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖)) |
12 | 3 | oveqd 7181 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → (𝑥(1st ‘𝑟)𝑦) = (𝑥𝐺𝑦)) |
13 | 12 | eleq1d 2817 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → ((𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖)) |
14 | 13 | ralbidv 3109 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖)) |
15 | | fveq2 6668 |
. . . . . . . . . . 11
⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = (2nd ‘𝑅)) |
16 | | idlval.2 |
. . . . . . . . . . 11
⊢ 𝐻 = (2nd ‘𝑅) |
17 | 15, 16 | eqtr4di 2791 |
. . . . . . . . . 10
⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = 𝐻) |
18 | 17 | oveqd 7181 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → (𝑧(2nd ‘𝑟)𝑥) = (𝑧𝐻𝑥)) |
19 | 18 | eleq1d 2817 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → ((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖)) |
20 | 17 | oveqd 7181 |
. . . . . . . . 9
⊢ (𝑟 = 𝑅 → (𝑥(2nd ‘𝑟)𝑧) = (𝑥𝐻𝑧)) |
21 | 20 | eleq1d 2817 |
. . . . . . . 8
⊢ (𝑟 = 𝑅 → ((𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖)) |
22 | 19, 21 | anbi12d 634 |
. . . . . . 7
⊢ (𝑟 = 𝑅 → (((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))) |
23 | 6, 22 | raleqbidv 3303 |
. . . . . 6
⊢ (𝑟 = 𝑅 → (∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))) |
24 | 14, 23 | anbi12d 634 |
. . . . 5
⊢ (𝑟 = 𝑅 → ((∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)) ↔ (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))) |
25 | 24 | ralbidv 3109 |
. . . 4
⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)) ↔ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))) |
26 | 11, 25 | anbi12d 634 |
. . 3
⊢ (𝑟 = 𝑅 → (((GId‘(1st
‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖))) ↔ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))) |
27 | 7, 26 | rabeqbidv 3386 |
. 2
⊢ (𝑟 = 𝑅 → {𝑖 ∈ 𝒫 ran (1st
‘𝑟) ∣
((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))} = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))}) |
28 | | df-idl 35780 |
. 2
⊢ Idl =
(𝑟 ∈ RingOps ↦
{𝑖 ∈ 𝒫 ran
(1st ‘𝑟)
∣ ((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))}) |
29 | 2 | fvexi 6682 |
. . . . . 6
⊢ 𝐺 ∈ V |
30 | 29 | rnex 7636 |
. . . . 5
⊢ ran 𝐺 ∈ V |
31 | 5, 30 | eqeltri 2829 |
. . . 4
⊢ 𝑋 ∈ V |
32 | 31 | pwex 5244 |
. . 3
⊢ 𝒫
𝑋 ∈ V |
33 | 32 | rabex 5197 |
. 2
⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V |
34 | 27, 28, 33 | fvmpt 6769 |
1
⊢ (𝑅 ∈ RingOps →
(Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))}) |