Theorem metuel2 23921

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 2829	. . 3 ⊢ (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷))
3	2	a1i 11	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷)))
4		metuel 23920	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉)))
5		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
6	5	imaeq2d 6013	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑑)))
7	6	cbvmptv 5218	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
8	7	elrnmpt 5911	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑))))
9	8	elv 3451	. . . . . . . . . 10 ⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)))
10	9	anbi1i 624	. . . . . . . . 9 ⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
11		r19.41v 3185	. . . . . . . . 9 ⊢ (∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
12	10, 11	bitr4i 277	. . . . . . . 8 ⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
13	12	exbii 1850	. . . . . . 7 ⊢ (∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
14		df-rex 3074	. . . . . . 7 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉))
15		rexcom4 3271	. . . . . . 7 ⊢ (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
16	13, 14, 15	3bitr4i 302	. . . . . 6 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
17		cnvexg 7861	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
18		imaexg 7852	. . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
19		sseq1 3969	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑑)) → (𝑤 ⊆ 𝑉 ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
20	19	ceqsexgv 3604	. . . . . . . . 9 ⊢ ((◡𝐷 “ (0[,)𝑑)) ∈ V → (∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
21	17, 18, 20	3syl 18	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
22	21	rexbidv 3175	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
23	22	adantr 481	. . . . . 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 282	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) → (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
25		cnvimass 6033	. . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
26		simpll 765	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
27		psmetf 23659	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
28		fdm 6677	. . . . . . . . . 10 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
29	26, 27, 28	3syl 18	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → dom 𝐷 = (𝑋 × 𝑋))
30	25, 29	sseqtrid 3996	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
31		ssrel2 5741	. . . . . . . 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 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)))
33		simplr 767	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
34		simpr 485	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
35	33, 34	opelxpd 5671	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
36	35	biantrurd 533	. . . . . . . . . . 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 23660	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
38	37	ad5ant145 1369	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
39	38	3biant1d 1478	. . . . . . . . . . . . 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 23665	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦))
41	40	biantrurd 533	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
42	41	ad5ant145 1369	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
43		0xr 11202	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ*
44		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ+)
45	44	rpxrd 12958	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ*)
46		elico1 13307	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ 𝑑 ∈ ℝ*) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
47	43, 45, 46	sylancr 587	. . . . . . . . . . . . 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 310	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝑥𝐷𝑦) ∈ (0[,)𝑑)))
49		df-ov 7360	. . . . . . . . . . . . 13 ⊢ (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
50	49	eleq1i 2828	. . . . . . . . . . . 12 ⊢ ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑))
51	48, 50	bitrdi 286	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (𝐷‘⟨𝑥, 𝑦⟩) ∈ (0[,)𝑑)))
52		simp-4l 781	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
53		ffn 6668	. . . . . . . . . . . 12 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋))
54		elpreima 7008	. . . . . . . . . . . 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 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))))
56	36, 51, 55	3bitr4d 310	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑))))
57	56	anasss 467	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑))))
58		df-br 5106	. . . . . . . . . 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 344	. . . . . . . 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 3210	. . . . . . 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 281	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → ((◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))
63	62	rexbidva 3173	. . . . 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 278	. . . 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 579	. . 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 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))))
67	3, 4, 66	3bitrd 304	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  ⟨cop 4592   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  0cc0 11051  ℝ^*cxr 11188   < clt 11189   ≤ cle 11190  ℝ⁺crp 12915  [,)cico 13266  PsMetcpsmet 20780  metUnifcmetu 20787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-2 12216  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-psmet 20788  df-fbas 20793  df-fg 20794  df-metu 20795

This theorem is referenced by: (None)