Step | Hyp | Ref
| Expression |
1 | | cfili 22874 |
. . . . . 6
⊢ (((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ∧ 𝑥 ∈ ℝ+) →
∃𝑢 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥) |
2 | 1 | adantll 746 |
. . . . 5
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ+) →
∃𝑢 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥) |
3 | | elfg 21485 |
. . . . . . . 8
⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑢 ∈ (𝑋filGen𝐵) ↔ (𝑢 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢))) |
4 | 3 | ad3antlr 763 |
. . . . . . 7
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ+) → (𝑢 ∈ (𝑋filGen𝐵) ↔ (𝑢 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢))) |
5 | | ssralv 3629 |
. . . . . . . . . . . . 13
⊢ (𝑦 ⊆ 𝑢 → (∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
6 | 5 | ralimdv 2946 |
. . . . . . . . . . . 12
⊢ (𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
7 | | ssralv 3629 |
. . . . . . . . . . . 12
⊢ (𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
8 | 6, 7 | syld 46 |
. . . . . . . . . . 11
⊢ (𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
9 | 8 | com12 32 |
. . . . . . . . . 10
⊢
(∀𝑧 ∈
𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → (𝑦 ⊆ 𝑢 → ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
10 | 9 | reximdv 2999 |
. . . . . . . . 9
⊢
(∀𝑧 ∈
𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → (∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
11 | 10 | com12 32 |
. . . . . . . 8
⊢
(∃𝑦 ∈
𝐵 𝑦 ⊆ 𝑢 → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
12 | 11 | adantl 481 |
. . . . . . 7
⊢ ((𝑢 ⊆ 𝑋 ∧ ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑢) → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
13 | 4, 12 | syl6bi 242 |
. . . . . 6
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ+) → (𝑢 ∈ (𝑋filGen𝐵) → (∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))) |
14 | 13 | rexlimdv 3012 |
. . . . 5
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ+) →
(∃𝑢 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑢 ∀𝑤 ∈ 𝑢 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
15 | 2, 14 | mpd 15 |
. . . 4
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥) |
16 | 15 | ralrimiva 2949 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) ∧ (𝑋filGen𝐵) ∈ (CauFil‘𝐷)) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥) |
17 | 16 | ex 449 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
18 | | ssfg 21486 |
. . . . . 6
⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵)) |
19 | 18 | adantl 481 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → 𝐵 ⊆ (𝑋filGen𝐵)) |
20 | | ssrexv 3630 |
. . . . . 6
⊢ (𝐵 ⊆ (𝑋filGen𝐵) → (∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
21 | 20 | ralimdv 2946 |
. . . . 5
⊢ (𝐵 ⊆ (𝑋filGen𝐵) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
22 | 19, 21 | syl 17 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |
23 | | fgcl 21492 |
. . . . 5
⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑋filGen𝐵) ∈ (Fil‘𝑋)) |
24 | 23 | adantl 481 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝑋filGen𝐵) ∈ (Fil‘𝑋)) |
25 | 22, 24 | jctild 564 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → ((𝑋filGen𝐵) ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))) |
26 | | iscfil2 22872 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ((𝑋filGen𝐵) ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))) |
27 | 26 | adantr 480 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ((𝑋filGen𝐵) ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ (𝑋filGen𝐵)∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))) |
28 | 25, 27 | sylibrd 248 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥 → (𝑋filGen𝐵) ∈ (CauFil‘𝐷))) |
29 | 17, 28 | impbid 201 |
1
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) |