metuel2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > metuel2		Structured version Visualization version GIF version

Theorem metuel2 24294

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 2825	. . 3 ⊢ (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷))
3	2	a1i 11	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ 𝑈 ↔ 𝑉 ∈ (metUnif‘𝐷)))
4		metuel 24293	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉)))
5		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → (0[,)𝑎) = (0[,)𝑑))
6	5	imaeq2d 6059	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → (^◡𝐷 “ (0[,)𝑎)) = (^◡𝐷 “ (0[,)𝑑)))
7	6	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) = (𝑑 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑑)))
8	7	elrnmpt 5955	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) ↔ ∃𝑑 ∈ ℝ⁺ 𝑤 = (^◡𝐷 “ (0[,)𝑑))))
9	8	elv 3480	. . . . . . . . . 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 3188	. . . . . . . . 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 3071	. . . . . . 7 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎)))𝑤 ⊆ 𝑉 ↔ ∃𝑤(𝑤 ∈ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) ∧ 𝑤 ⊆ 𝑉))
15		rexcom4 3285	. . . . . . 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 7917	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ^◡𝐷 ∈ V)
18		imaexg 7908	. . . . . . . . 9 ⊢ (^◡𝐷 ∈ V → (^◡𝐷 “ (0[,)𝑑)) ∈ V)
19		sseq1 4007	. . . . . . . . . 10 ⊢ (𝑤 = (^◡𝐷 “ (0[,)𝑑)) → (𝑤 ⊆ 𝑉 ↔ (^◡𝐷 “ (0[,)𝑑)) ⊆ 𝑉))
20	19	ceqsexgv 3642	. . . . . . . . 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 3178	. . . . . . 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 6080	. . . . . . . . 9 ⊢ (^◡𝐷 “ (0[,)𝑑)) ⊆ dom 𝐷
26		simpll 765	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ⁺) → 𝐷 ∈ (PsMet‘𝑋))
27		psmetf 24032	. . . . . . . . . 10 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ^*)
28		fdm 6726	. . . . . . . . . 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 4034	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ⁺) → (^◡𝐷 “ (0[,)𝑑)) ⊆ (𝑋 × 𝑋))
31		ssrel2 5785	. . . . . . . 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 5715	. . . . . . . . . . . 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 24033	. . . . . . . . . . . . . . 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 24038	. . . . . . . . . . . . . . 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 11265	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ^*
44		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ⁺)
45	44	rpxrd 13021	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ⊆ (𝑋 × 𝑋)) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑑 ∈ ℝ^*)
46		elico1 13371	. . . . . . . . . . . . . 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 7414	. . . . . . . . . . . . 13 ⊢ (𝑥𝐷𝑦) = (𝐷‘⟨𝑥, 𝑦⟩)
50	49	eleq1i 2824	. . . . . . . . . . . 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 6717	. . . . . . . . . . . 12 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ^* → 𝐷 Fn (𝑋 × 𝑋))
54		elpreima 7059	. . . . . . . . . . . 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 5149	. . . . . . . . . 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 3216	. . . . . . 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 3176	. . . . 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))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 0cc0 11112 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 ℝ⁺crp 12978 [,)cico 13330 PsMetcpsmet 21128 metUnifcmetu 21135

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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ico 13334 df-psmet 21136 df-fbas 21141 df-fg 21142 df-metu 21143

This theorem is referenced by: (None)

W3C validator