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Theorem metuel2 22280
Description: Elementhood in the uniform structure generated by a metric 𝐷 (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metuel2.u 𝑈 = (metUnif‘𝐷)
Assertion
Ref Expression
metuel2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
Distinct variable groups:   𝑥,𝑑,𝑦,𝐷   𝑉,𝑑,𝑥,𝑦   𝑋,𝑑,𝑥,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦,𝑑)

Proof of Theorem metuel2
Dummy variables 𝑎 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metuel2.u . . . 4 𝑈 = (metUnif‘𝐷)
21eleq2i 2690 . . 3 (𝑉𝑈𝑉 ∈ (metUnif‘𝐷))
32a1i 11 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈𝑉 ∈ (metUnif‘𝐷)))
4 metuel 22279 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉)))
5 vex 3189 . . . . . . . . . . 11 𝑤 ∈ V
6 oveq2 6612 . . . . . . . . . . . . . 14 (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
76imaeq2d 5425 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑑)))
87cbvmptv 4710 . . . . . . . . . . . 12 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
98elrnmpt 5332 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑))))
105, 9ax-mp 5 . . . . . . . . . 10 (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)))
1110anbi1i 730 . . . . . . . . 9 ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
12 r19.41v 3081 . . . . . . . . 9 (∃𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1311, 12bitr4i 267 . . . . . . . 8 ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1413exbii 1771 . . . . . . 7 (∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ ∃𝑤𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
15 df-rex 2913 . . . . . . 7 (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉))
16 rexcom4 3211 . . . . . . 7 (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑤𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1714, 15, 163bitr4i 292 . . . . . 6 (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
18 cnvexg 7059 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
19 imaexg 7050 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑑)) ∈ V)
20 sseq1 3605 . . . . . . . . . 10 (𝑤 = (𝐷 “ (0[,)𝑑)) → (𝑤𝑉 ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2120ceqsexgv 3318 . . . . . . . . 9 ((𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2218, 19, 213syl 18 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2322rexbidv 3045 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2423adantr 481 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2517, 24syl5bb 272 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
26 cnvimass 5444 . . . . . . . . 9 (𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
27 simpll 789 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
28 psmetf 22021 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
29 fdm 6008 . . . . . . . . . 10 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
3027, 28, 293syl 18 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
3126, 30syl5sseq 3632 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
32 ssrel2 5171 . . . . . . . 8 ((𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
3331, 32syl 17 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
34 simplr 791 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥𝑋)
35 simpr 477 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
36 opelxp 5106 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ↔ (𝑥𝑋𝑦𝑋))
3734, 35, 36sylanbrc 697 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
3837biantrurd 529 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))))
39 simp-4l 805 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝐷 ∈ (PsMet‘𝑋))
40 psmetcl 22022 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
4139, 34, 35, 40syl3anc 1323 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
42413biant1d 1438 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
43 psmetge0 22027 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → 0 ≤ (𝑥𝐷𝑦))
4443biantrurd 529 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
4539, 34, 35, 44syl3anc 1323 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
46 0xr 10030 . . . . . . . . . . . . . 14 0 ∈ ℝ*
47 simpllr 798 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑑 ∈ ℝ+)
4847rpxrd 11817 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑑 ∈ ℝ*)
49 elico1 12160 . . . . . . . . . . . . . 14 ((0 ∈ ℝ*𝑑 ∈ ℝ*) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
5046, 48, 49sylancr 694 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
5142, 45, 503bitr4d 300 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝑥𝐷𝑦) ∈ (0[,)𝑑)))
52 df-ov 6607 . . . . . . . . . . . . 13 (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
5352eleq1i 2689 . . . . . . . . . . . 12 ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))
5451, 53syl6bb 276 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑)))
55 ffn 6002 . . . . . . . . . . . 12 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
56 elpreima 6293 . . . . . . . . . . . 12 (𝐷 Fn (𝑋 × 𝑋) → (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))))
5739, 28, 55, 564syl 19 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))))
5838, 54, 573bitr4d 300 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑))))
5958anasss 678 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑))))
60 df-br 4614 . . . . . . . . . 10 (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉)
6160a1i 11 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉))
6259, 61imbi12d 334 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
63622ralbidva 2982 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
6433, 63bitr4d 271 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6564rexbidva 3042 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6625, 65bitrd 268 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6766pm5.32da 672 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ((𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
6867adantl 482 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
693, 4, 683bitrd 294 1 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  wss 3555  c0 3891  cop 4154   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  0cc0 9880  *cxr 10017   < clt 10018  cle 10019  +crp 11776  [,)cico 12119  PsMetcpsmet 19649  metUnifcmetu 19656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-2 11023  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ico 12123  df-psmet 19657  df-fbas 19662  df-fg 19663  df-metu 19664
This theorem is referenced by: (None)
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