Step | Hyp | Ref
| Expression |
1 | | simpr3 1198 |
. . 3
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) |
2 | | dfiin2g 4941 |
. . 3
⊢
(∀𝑥 ∈
𝐴 𝐵 ∈ 𝐽 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
3 | 1, 2 | syl 17 |
. 2
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
4 | | simpl 486 |
. . 3
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → 𝐽 ∈ Top) |
5 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
6 | 5 | rnmpt 5824 |
. . . 4
⊢ ran
(𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
7 | 5 | fmpt 6927 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 𝐵 ∈ 𝐽 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐽) |
8 | 1, 7 | sylib 221 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐽) |
9 | 8 | frnd 6553 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐽) |
10 | 6, 9 | eqsstrrid 3950 |
. . 3
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽) |
11 | 8 | fdmd 6556 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
12 | | simpr2 1197 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → 𝐴 ≠ ∅) |
13 | 11, 12 | eqnetrd 3008 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) |
14 | | dm0rn0 5794 |
. . . . . 6
⊢ (dom
(𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∅) |
15 | 6 | eqeq1i 2742 |
. . . . . 6
⊢ (ran
(𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = ∅) |
16 | 14, 15 | bitri 278 |
. . . . 5
⊢ (dom
(𝑥 ∈ 𝐴 ↦ 𝐵) = ∅ ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = ∅) |
17 | 16 | necon3bii 2993 |
. . . 4
⊢ (dom
(𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅) |
18 | 13, 17 | sylib 221 |
. . 3
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅) |
19 | | simpr1 1196 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → 𝐴 ∈ Fin) |
20 | | abrexfi 8976 |
. . . 4
⊢ (𝐴 ∈ Fin → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ Fin) |
21 | 19, 20 | syl 17 |
. . 3
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ Fin) |
22 | | fiinopn 21798 |
. . . 4
⊢ (𝐽 ∈ Top → (({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽 ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ Fin) → ∩ {𝑦
∣ ∃𝑥 ∈
𝐴 𝑦 = 𝐵} ∈ 𝐽)) |
23 | 22 | imp 410 |
. . 3
⊢ ((𝐽 ∈ Top ∧ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐽 ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ∧ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ Fin)) → ∩ {𝑦
∣ ∃𝑥 ∈
𝐴 𝑦 = 𝐵} ∈ 𝐽) |
24 | 4, 10, 18, 21, 23 | syl13anc 1374 |
. 2
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ 𝐽) |
25 | 3, 24 | eqeltrd 2838 |
1
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧
∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) |