Theorem metuel2 24751

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 2857	. . 3 ⊢ (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷))
3	2	a1i 11	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷)))
4		metuel 24750	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉)))
5		oveq2 7424	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
6	5	imaeq2d 6064	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑑)))
7	6	cbvmptv 5217	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑑)))
8	7	elrnmpt 5950	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑))))
9	8	elv 3462	. . . . . . . . . 10 ⊢ (𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)))
10	9	anbi1i 636	. . . . . . . . 9 ⊢ ((𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ (∃𝑑 ∈ ℝ+ 𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
11		r19.41v 3197	. . . . . . . . 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 1881	. . . . . . 7 ⊢ (∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑤∃𝑑 ∈ ℝ+ (𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉))
14		df-rex 3092	. . . . . . 7 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉))
15		rexcom4 3294	. . . . . . 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 7923	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ◡𝐷 ∈ V)
18		imaexg 7912	. . . . . . . . 9 ⊢ (◡𝐷 ∈ V → (◡𝐷 “ (0[,)𝑑)) ∈ V)
19		sseq1 3963	. . . . . . . . . 10 ⊢ (𝑤 = (◡𝐷 “ (0[,)𝑑)) → (𝑤 ⊆ 𝑉 ↔ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
20	19	ceqsexgv 3615	. . . . . . . . 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 3191	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (∃𝑑 ∈ ℝ+ ∃𝑤(𝑤 = (◡𝐷 “ (0[,)𝑑)) ∧ 𝑤 ⊆ 𝑉) ↔ ∃𝑑 ∈ ℝ+ (◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
23	22	adantr 486	. . . . . 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 6086	. . . . . . . . 9 ⊢ (◡𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
26		simpll 779	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → 𝐷 ∈ (PsMet‘𝑋))
27		psmetf 24492	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
28		fdm 6719	. . . . . . . . . 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 3980	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) → (◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
31		ssrel2 5773	. . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ 𝑋)
34		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
35	33, 34	opelxpd 5702	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑋))
36	35	biantrurd 542	. . . . . . . . . . 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 24493	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
38	37	ad5ant145 1396	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝐷𝑦) ∈ ℝ*)
39	38	3biant1d 1509	. . . . . . . . . . . . 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 24498	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑥𝐷𝑦))
41	40	biantrurd 542	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
42	41	ad5ant145 1396	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) < 𝑑 ↔ (0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
43		0xr 11267	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ*
44		simpllr 788	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ+)
45	44	rpxrd 13072	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ*)
46		elico1 13426	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ 𝑑 ∈ ℝ*) → ((𝑥𝐷𝑦) ∈ (0[,)𝑑) ↔ ((𝑥𝐷𝑦) ∈ ℝ* ∧ 0 ≤ (𝑥𝐷𝑦) ∧ (𝑥𝐷𝑦) < 𝑑)))
47	43, 45, 46	sylancr 599	. . . . . . . . . . . . 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 7419	. . . . . . . . . . . . 13 ⊢ (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
50	49	eleq1i 2856	. . . . . . . . . . . 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 795	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝐷 ∈ (PsMet‘𝑋))
53		ffn 6709	. . . . . . . . . . . 12 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → 𝐷 Fn (𝑋 × 𝑋))
54		elpreima 7057	. . . . . . . . . . . 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 472	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ+) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) < 𝑑 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐷 “ (0[,)𝑑))))
58		df-br 5112	. . . . . . . . . 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 3229	. . . . . . 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 3189	. . . . 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 590	. . 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 487	. 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  0cc0 11111  ℝ^*cxr 11253   < clt 11254   ≤ cle 11255  ℝ⁺crp 13027  [,)cico 13385  PsMetcpsmet 21535  metUnifcmetu 21542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-rp 13028  df-xneg 13148  df-xadd 13149  df-xmul 13150  df-ico 13389  df-psmet 21543  df-fbas 21548  df-fg 21549  df-metu 21550

This theorem is used by: (None)