Step | Hyp | Ref
| Expression |
1 | | metuel2.u |
. . . 4
⊢ 𝑈 = (metUnif‘𝐷) |
2 | 1 | eleq2i 2693 |
. . 3
⊢ (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷)) |
3 | 2 | a1i 11 |
. 2
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷))) |
4 | | metuel 22369 |
. 2
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉))) |
5 | | vex 3203 |
. . . . . . . . . . 11
⊢ 𝑤 ∈ V |
6 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑)) |
7 | 6 | imaeq2d 5466 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑑 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑑))) |
8 | 7 | cbvmptv 4750 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ ℝ+
↦ (◡𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑))) |
9 | 8 | elrnmpt 5372 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)))) |
10 | 5, 9 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑))) |
11 | 10 | anbi1i 731 |
. . . . . . . . 9
⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉)) |
12 | | r19.41v 3089 |
. . . . . . . . 9
⊢
(∃𝑑 ∈
ℝ+ (𝑤 =
(◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉)) |
13 | 11, 12 | bitr4i 267 |
. . . . . . . 8
⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉)) |
14 | 13 | exbii 1774 |
. . . . . . 7
⊢
(∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉)) |
15 | | df-rex 2918 |
. . . . . . 7
⊢
(∃𝑤 ∈ ran
(𝑎 ∈
ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉)) |
16 | | rexcom4 3225 |
. . . . . . 7
⊢
(∃𝑑 ∈
ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉)) |
17 | 14, 15, 16 | 3bitr4i 292 |
. . . . . 6
⊢
(∃𝑤 ∈ ran
(𝑎 ∈
ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉)) |
18 | | cnvexg 7112 |
. . . . . . . . 9
⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V) |
19 | | imaexg 7103 |
. . . . . . . . 9
⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V) |
20 | | sseq1 3626 |
. . . . . . . . . 10
⊢ (𝑤 = (◡𝐷 “ (0[,)𝑑)) → (𝑤 ⊆ 𝑉 ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉)) |
21 | 20 | ceqsexgv 3335 |
. . . . . . . . 9
⊢ ((◡𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉)) |
22 | 18, 19, 21 | 3syl 18 |
. . . . . . . 8
⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉)) |
23 | 22 | rexbidv 3052 |
. . . . . . 7
⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+
∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉)) |
24 | 23 | adantr 481 |
. . . . . 6
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉)) |
25 | 17, 24 | syl5bb 272 |
. . . . 5
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉)) |
26 | | cnvimass 5485 |
. . . . . . . . 9
⊢ (◡𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷 |
27 | | simpll 790 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋)) |
28 | | psmetf 22111 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
29 | | fdm 6051 |
. . . . . . . . . 10
⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom
𝐷 = (𝑋 × 𝑋)) |
30 | 27, 28, 29 | 3syl 18 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋)) |
31 | 26, 30 | syl5sseq 3653 |
. . . . . . . 8
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋)) |
32 | | ssrel2 5210 |
. . . . . . . 8
⊢ ((◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋) → ((◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)) → 〈𝑥, 𝑦〉 ∈ 𝑉))) |
33 | 31, 32 | syl 17 |
. . . . . . 7
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)) → 〈𝑥, 𝑦〉 ∈ 𝑉))) |
34 | | simplr 792 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
35 | | simpr 477 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋) |
36 | | opelxp 5146 |
. . . . . . . . . . . . 13
⊢
(〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) |
37 | 34, 35, 36 | sylanbrc 698 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑋)) |
38 | 37 | biantrurd 529 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐷‘〈𝑥, 𝑦〉) ∈ (0[,)𝑑) ↔ (〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝑥, 𝑦〉) ∈ (0[,)𝑑)))) |
39 | | simp-4l 806 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋)) |
40 | | psmetcl 22112 |
. . . . . . . . . . . . . . 15
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈
ℝ*) |
41 | 39, 34, 35, 40 | syl3anc 1326 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈
ℝ*) |
42 | 41 | 3biant1d 1441 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤
(𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑))) |
43 | | psmetge0 22117 |
. . . . . . . . . . . . . . 15
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦)) |
44 | 43 | biantrurd 529 |
. . . . . . . . . . . . . 14
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑))) |
45 | 39, 34, 35, 44 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑))) |
46 | | 0xr 10086 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ* |
47 | | simpllr 799 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ+) |
48 | 47 | rpxrd 11873 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ*) |
49 | | elico1 12218 |
. . . . . . . . . . . . . 14
⊢ ((0
∈ ℝ* ∧ 𝑑 ∈ ℝ*) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤
(𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑))) |
50 | 46, 48, 49 | sylancr 695 |
. . . . . . . . . . . . 13
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤
(𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑))) |
51 | 42, 45, 50 | 3bitr4d 300 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝑥𝐷𝑦) ∈ (0[,)𝑑))) |
52 | | df-ov 6653 |
. . . . . . . . . . . . 13
⊢ (𝑥𝐷𝑦) = (𝐷‘〈𝑥, 𝑦〉) |
53 | 52 | eleq1i 2692 |
. . . . . . . . . . . 12
⊢ ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ (𝐷‘〈𝑥, 𝑦〉) ∈ (0[,)𝑑)) |
54 | 51, 53 | syl6bb 276 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝐷‘〈𝑥, 𝑦〉) ∈ (0[,)𝑑))) |
55 | | ffn 6045 |
. . . . . . . . . . . 12
⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋)) |
56 | | elpreima 6337 |
. . . . . . . . . . . 12
⊢ (𝐷 Fn (𝑋 × 𝑋) → (〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)) ↔ (〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝑥, 𝑦〉) ∈ (0[,)𝑑)))) |
57 | 39, 28, 55, 56 | 4syl 19 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)) ↔ (〈𝑥, 𝑦〉 ∈ (𝑋 × 𝑋) ∧ (𝐷‘〈𝑥, 𝑦〉) ∈ (0[,)𝑑)))) |
58 | 38, 54, 57 | 3bitr4d 300 |
. . . . . . . . . 10
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)))) |
59 | 58 | anasss 679 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ 〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)))) |
60 | | df-br 4654 |
. . . . . . . . . 10
⊢ (𝑥𝑉𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑉) |
61 | 60 | a1i 11 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑉𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑉)) |
62 | 59, 61 | imbi12d 334 |
. . . . . . . 8
⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦) ↔ (〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)) → 〈𝑥, 𝑦〉 ∈ 𝑉))) |
63 | 62 | 2ralbidva 2988 |
. . . . . . 7
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) →
(∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (〈𝑥, 𝑦〉 ∈ (◡𝐷 “ (0[,)𝑑)) → 〈𝑥, 𝑦〉 ∈ 𝑉))) |
64 | 33, 63 | bitr4d 271 |
. . . . . 6
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))) |
65 | 64 | rexbidva 3049 |
. . . . 5
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))) |
66 | 25, 65 | bitrd 268 |
. . . 4
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))) |
67 | 66 | pm5.32da 673 |
. . 3
⊢ (𝐷 ∈ (PsMet‘𝑋) → ((𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))) |
68 | 67 | adantl 482 |
. 2
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))) |
69 | 3, 4, 68 | 3bitrd 294 |
1
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))) |