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Theorem metuel2 23090
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 2909 . . 3 (𝑉𝑈𝑉 ∈ (metUnif‘𝐷))
32a1i 11 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈𝑉 ∈ (metUnif‘𝐷)))
4 metuel 23089 . 2 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉)))
5 oveq2 7156 . . . . . . . . . . . . . 14 (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
65imaeq2d 5927 . . . . . . . . . . . . 13 (𝑎 = 𝑑 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑑)))
76cbvmptv 5166 . . . . . . . . . . . 12 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑑)))
87elrnmpt 5827 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑))))
98elv 3505 . . . . . . . . . 10 (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)))
109anbi1i 623 . . . . . . . . 9 ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
11 r19.41v 3352 . . . . . . . . 9 (∃𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1210, 11bitr4i 279 . . . . . . . 8 ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1312exbii 1841 . . . . . . 7 (∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉) ↔ ∃𝑤𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
14 df-rex 3149 . . . . . . 7 (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) ∧ 𝑤𝑉))
15 rexcom4 3254 . . . . . . 7 (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑤𝑑 ∈ ℝ+ (𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
1613, 14, 153bitr4i 304 . . . . . 6 (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉))
17 cnvexg 7617 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
18 imaexg 7608 . . . . . . . . 9 (𝐷 ∈ V → (𝐷 “ (0[,)𝑑)) ∈ V)
19 sseq1 3996 . . . . . . . . . 10 (𝑤 = (𝐷 “ (0[,)𝑑)) → (𝑤𝑉 ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2019ceqsexgv 3651 . . . . . . . . 9 ((𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2117, 18, 203syl 18 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2221rexbidv 3302 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2322adantr 481 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+𝑤(𝑤 = (𝐷 “ (0[,)𝑑)) ∧ 𝑤𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
2416, 23syl5bb 284 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
25 cnvimass 5947 . . . . . . . . 9 (𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
26 simpll 763 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
27 psmetf 22831 . . . . . . . . . 10 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
28 fdm 6519 . . . . . . . . . 10 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
2926, 27, 283syl 18 . . . . . . . . 9 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
3025, 29sseqtrid 4023 . . . . . . . 8 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
31 ssrel2 5658 . . . . . . . 8 ((𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
3230, 31syl 17 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
33 simplr 765 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑥𝑋)
34 simpr 485 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑦𝑋)
3533, 34opelxpd 5592 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
3635biantrurd 533 . . . . . . . . . . 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))))
37 psmetcl 22832 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
3837ad5ant145 1363 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
39383biant1d 1471 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
40 psmetge0 22837 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → 0 ≤ (𝑥𝐷𝑦))
4140biantrurd 533 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥𝑋𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
4241ad5ant145 1363 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
43 0xr 10677 . . . . . . . . . . . . . 14 0 ∈ ℝ*
44 simpllr 772 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑑 ∈ ℝ+)
4544rpxrd 12422 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝑑 ∈ ℝ*)
46 elico1 12771 . . . . . . . . . . . . . 14 ((0 ∈ ℝ*𝑑 ∈ ℝ*) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
4743, 45, 46sylancr 587 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
4839, 42, 473bitr4d 312 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝑥𝐷𝑦) ∈ (0[,)𝑑)))
49 df-ov 7151 . . . . . . . . . . . . 13 (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
5049eleq1i 2908 . . . . . . . . . . . 12 ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))
5148, 50syl6bb 288 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑)))
52 simp-4l 779 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → 𝐷 ∈ (PsMet‘𝑋))
53 ffn 6511 . . . . . . . . . . . 12 (𝐷:(𝑋 × 𝑋)⟶ℝ*𝐷 Fn (𝑋 × 𝑋))
54 elpreima 6824 . . . . . . . . . . . 12 (𝐷 Fn (𝑋 × 𝑋) → (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))))
5552, 27, 53, 544syl 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))))
5636, 51, 553bitr4d 312 . . . . . . . . . 10 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑦𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑))))
5756anasss 467 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑))))
58 df-br 5064 . . . . . . . . . 10 (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉)
5958a1i 11 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉))
6057, 59imbi12d 346 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥𝑋𝑦𝑋)) → (((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
61602ralbidva 3203 . . . . . . 7 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦) ↔ ∀𝑥𝑋𝑦𝑋 (⟨𝑥, 𝑦⟩ ∈ (𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
6232, 61bitr4d 283 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6362rexbidva 3301 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ (𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6424, 63bitrd 280 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉 ↔ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦)))
6564pm5.32da 579 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ((𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
6665adantl 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))))
673, 4, 663bitrd 306 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 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  wss 3940  c0 4295  cop 4570   class class class wbr 5063  cmpt 5143   × cxp 5552  ccnv 5553  dom cdm 5554  ran crn 5555  cima 5557   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  0cc0 10526  *cxr 10663   < clt 10664  cle 10665  +crp 12379  [,)cico 12730  PsMetcpsmet 20445  metUnifcmetu 20452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-2 11689  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-ico 12734  df-psmet 20453  df-fbas 20458  df-fg 20459  df-metu 20460
This theorem is referenced by: (None)
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