Step | Hyp | Ref
| Expression |
1 | | elrnust 23122 |
. . . . 5
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran
UnifOn) |
2 | | unieq 4830 |
. . . . . . . . 9
⊢ (𝑢 = 𝑈 → ∪ 𝑢 = ∪
𝑈) |
3 | 2 | dmeqd 5774 |
. . . . . . . 8
⊢ (𝑢 = 𝑈 → dom ∪
𝑢 = dom ∪ 𝑈) |
4 | 3 | fveq2d 6721 |
. . . . . . 7
⊢ (𝑢 = 𝑈 → (fBas‘dom ∪ 𝑢) =
(fBas‘dom ∪ 𝑈)) |
5 | | raleq 3319 |
. . . . . . 7
⊢ (𝑢 = 𝑈 → (∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣)) |
6 | 4, 5 | rabeqbidv 3396 |
. . . . . 6
⊢ (𝑢 = 𝑈 → {𝑓 ∈ (fBas‘dom ∪ 𝑢)
∣ ∀𝑣 ∈
𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} = {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) |
7 | | df-cfilu 23184 |
. . . . . 6
⊢
CauFilu = (𝑢 ∈ ∪ ran
UnifOn ↦ {𝑓 ∈
(fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) |
8 | | fvex 6730 |
. . . . . . 7
⊢
(fBas‘dom ∪ 𝑈) ∈ V |
9 | 8 | rabex 5225 |
. . . . . 6
⊢ {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ∈ V |
10 | 6, 7, 9 | fvmpt 6818 |
. . . . 5
⊢ (𝑈 ∈ ∪ ran UnifOn → (CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) |
11 | 1, 10 | syl 17 |
. . . 4
⊢ (𝑈 ∈ (UnifOn‘𝑋) →
(CauFilu‘𝑈) = {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) |
12 | 11 | eleq2d 2823 |
. . 3
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ 𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})) |
13 | | rexeq 3320 |
. . . . 5
⊢ (𝑓 = 𝐹 → (∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) |
14 | 13 | ralbidv 3118 |
. . . 4
⊢ (𝑓 = 𝐹 → (∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣 ↔ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) |
15 | 14 | elrab 3602 |
. . 3
⊢ (𝐹 ∈ {𝑓 ∈ (fBas‘dom ∪ 𝑈)
∣ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣} ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈)
∧ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) |
16 | 12, 15 | bitrdi 290 |
. 2
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈)
∧ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) |
17 | | ustbas2 23123 |
. . . . 5
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈) |
18 | 17 | fveq2d 6721 |
. . . 4
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (fBas‘𝑋) = (fBas‘dom ∪ 𝑈)) |
19 | 18 | eleq2d 2823 |
. . 3
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ 𝐹 ∈ (fBas‘dom ∪ 𝑈))) |
20 | 19 | anbi1d 633 |
. 2
⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣) ↔ (𝐹 ∈ (fBas‘dom ∪ 𝑈)
∧ ∀𝑣 ∈
𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) |
21 | 16, 20 | bitr4d 285 |
1
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) |