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Theorem metuel2 22180
 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 2680 . . 3 (𝑉𝑈𝑉 ∈ (metUnif‘𝐷))
32a1i 11 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈𝑉 ∈ (metUnif‘𝐷)))
4 metuel 22179 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉)))
5 vex 3176 . . . . . . . . . . 11 𝑤 ∈ V
6 oveq2 6557 . . . . . . . . . . . . . 14 (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
76imaeq2d 5385 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑑)))
87cbvmptv 4678 . . . . . . . . . . . 12 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
98elrnmpt 5293 . . . . . . . . . . 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 727 . . . . . . . . 9 ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
12 r19.41v 3070 . . . . . . . . 9 (∃𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1311, 12bitr4i 266 . . . . . . . 8 ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1413exbii 1764 . . . . . . 7 (∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ ∃𝑤𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
15 df-rex 2902 . . . . . . 7 (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉))
16 rexcom4 3198 . . . . . . 7 (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑤𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1714, 15, 163bitr4i 291 . . . . . 6 (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
18 cnvexg 7005 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
19 imaexg 6995 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑑)) ∈ V)
20 sseq1 3589 . . . . . . . . . 10 (𝑤 = (𝐷 “ (0[,)𝑑)) → (𝑤𝑉 ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2120ceqsexgv 3305 . . . . . . . . 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 3034 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2423adantr 480 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2517, 24syl5bb 271 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
26 cnvimass 5404 . . . . . . . . 9 (𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
27 simpll 786 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
28 psmetf 21921 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
29 fdm 5964 . . . . . . . . . 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 3616 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
32 ssrel2 5133 . . . . . . . 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 788 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥𝑋)
35 simpr 476 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
36 opelxp 5070 . . . . . . . . . . . . 13 (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ↔ (𝑥𝑋𝑦𝑋))
3734, 35, 36sylanbrc 695 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
3837biantrurd 528 . . . . . . . . . . 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 802 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝐷 ∈ (PsMet‘𝑋))
40 psmetcl 21922 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
4139, 34, 35, 40syl3anc 1318 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
42413biant1d 1433 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
43 psmetge0 21927 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → 0 ≤ (𝑥𝐷𝑦))
4443biantrurd 528 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
4539, 34, 35, 44syl3anc 1318 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
46 0xr 9965 . . . . . . . . . . . . . 14 0 ∈ ℝ*
47 simpllr 795 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑑 ∈ ℝ+)
4847rpxrd 11749 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑑 ∈ ℝ*)
49 elico1 12089 . . . . . . . . . . . . . 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 299 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝑥𝐷𝑦) ∈ (0[,)𝑑)))
52 df-ov 6552 . . . . . . . . . . . . 13 (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
5352eleq1i 2679 . . . . . . . . . . . 12 ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))
5451, 53syl6bb 275 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑)))
55 ffn 5958 . . . . . . . . . . . 12 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
56 elpreima 6245 . . . . . . . . . . . 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 299 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑))))
5958anasss 677 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑))))
60 df-br 4584 . . . . . . . . . 10 (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉)
6160a1i 11 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉))
6259, 61imbi12d 333 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
63622ralbidva 2971 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
6433, 63bitr4d 270 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6564rexbidva 3031 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6625, 65bitrd 267 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6766pm5.32da 671 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ((𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
6867adantl 481 . 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 293 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  ℝ+crp 11708  [,)cico 12048  PsMetcpsmet 19551  metUnifcmetu 19558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-2 10956  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-psmet 19559  df-fbas 19564  df-fg 19565  df-metu 19566 This theorem is referenced by: (None)
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