Theorem metuel2 24687

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)) → (𝑉 ∈ 𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))))

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.

Step	Hyp	Ref	Expression
1		metuel2.u	. . . 4 ⊢ 𝑈 = (metUnif‘𝐷)
2	1	eleq2i 2861	. . 3 ⊢ (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷))
3	2	a1i 11	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷)))
4		metuel 24686	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉)))
5		oveq2 7416	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
6	5	imaeq2d 6060	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑑)))
7	6	cbvmptv 5216	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
8	7	elrnmpt 5946	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑))))
9	8	elv 3468	. . . . . . . . . 10 ⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)))
10	9	anbi1i 635	. . . . . . . . 9 ⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
11		r19.41v 3201	. . . . . . . . 9 ⊢ (∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
12	10, 11	bitr4i 281	. . . . . . . 8 ⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
13	12	exbii 1875	. . . . . . 7 ⊢ (∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
14		df-rex 3096	. . . . . . 7 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉))
15		rexcom4 3298	. . . . . . 7 ⊢ (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
16	13, 14, 15	3bitr4i 306	. . . . . 6 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
17		cnvexg 7917	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
18		imaexg 7906	. . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
19		sseq1 3970	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑑)) → (𝑤 ⊆ 𝑉 ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
20	19	ceqsexgv 3622	. . . . . . . . 9 ⊢ ((◡𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
21	17, 18, 20	3syl 19	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
22	21	rexbidv 3195	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
23	22	adantr 485	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
24	16, 23	bitrid 286	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
25		cnvimass 6082	. . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
26		simpll 778	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
27		psmetf 24428	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
28		fdm 6713	. . . . . . . . . 10 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
29	26, 27, 28	3syl 19	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
30	25, 29	sseqtrid 3987	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
31		ssrel2 5769	. . . . . . . 8 ⊢ ((◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋) → ((◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
32	30, 31	syl 18	. . . . . . 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)))
33		simplr 780	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
34		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
35	33, 34	opelxpd 5698	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
36	35	biantrurd 541	. . . . . . . . . . 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))))
37		psmetcl 24429	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
38	37	ad5ant145 1394	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
39	38	3biant1d 1506	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
40		psmetge0 24434	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦))
41	40	biantrurd 541	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
42	41	ad5ant145 1394	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
43		0xr 11252	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ*
44		simpllr 787	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ+)
45	44	rpxrd 13057	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ*)
46		elico1 13411	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ 𝑑 ∈ ℝ*) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
47	43, 45, 46	sylancr 598	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
48	39, 42, 47	3bitr4d 314	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝑥𝐷𝑦) ∈ (0[,)𝑑)))
49		df-ov 7411	. . . . . . . . . . . . 13 ⊢ (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
50	49	eleq1i 2860	. . . . . . . . . . . 12 ⊢ ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))
51	48, 50	bitrdi 290	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑)))
52		simp-4l 794	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
53		ffn 6703	. . . . . . . . . . . 12 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋))
54		elpreima 7051	. . . . . . . . . . . 12 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋) ∧ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))))
55	52, 27, 53, 54	4syl 20	. . . . . . . . . . 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))))
56	36, 51, 55	3bitr4d 314	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑))))
57	56	anasss 471	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑))))
58		df-br 5111	. . . . . . . . . 10 ⊢ (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉)
59	58	a1i 11	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑉𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑉))
60	57, 59	imbi12d 347	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑)) → ⟨𝑥, 𝑦⟩ ∈ 𝑉)))
61	60	2ralbidva 3233	. . . . . . 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)))
62	32, 61	bitr4d 285	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))
63	62	rexbidva 3193	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))
64	24, 63	bitrd 282	. . . 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)))
65	64	pm5.32da 589	. . 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))))
66	65	adantl 486	. 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))))
67	3, 4, 66	3bitrd 308	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ⟨cop 4597   class class class wbr 5110   ↦ cmpt 5193   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   “ cima 5662   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  0cc0 11096  ℝ^*cxr 11238   < clt 11239   ≤ cle 11240  ℝ⁺crp 13012  [,)cico 13370  PsMetcpsmet 21471  metUnifcmetu 21478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-nn 12230  df-2 12299  df-rp 13013  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ico 13374  df-psmet 21479  df-fbas 21484  df-fg 21485  df-metu 21486

This theorem is referenced by: (None)