ssbnd - Mathbox for Jeff Madsen

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Theorem ssbnd 35226

Description: A subset of a metric space is bounded iff it is contained in a ball around 𝑃, for any 𝑃 in the larger space. (Contributed by Mario Carneiro, 14-Sep-2015.)

**Hypothesis**
Ref	Expression
ssbnd.2	⊢ 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))

**Assertion**
Ref	Expression
ssbnd	⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ (Bnd‘𝑌) ↔ ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))

Distinct variable groups: 𝑀,𝑑 𝑁,𝑑 𝑃,𝑑 𝑋,𝑑 𝑌,𝑑

Proof of Theorem ssbnd Dummy variables 𝑟 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 10632	. . . . . . 7 ⊢ 0 ∈ ℝ
2	1	ne0ii 4253	. . . . . 6 ⊢ ℝ ≠ ∅
3		0ss 4304	. . . . . . . 8 ⊢ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4		sseq1 3940	. . . . . . . 8 ⊢ (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
5	3, 4	mpbiri 261	. . . . . . 7 ⊢ (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6	5	ralrimivw 3150	. . . . . 6 ⊢ (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7		r19.2z 4398	. . . . . 6 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
8	2, 6, 7	sylancr 590	. . . . 5 ⊢ (𝑌 = ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
9	8	a1i 11	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 = ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
10		isbnd2 35221	. . . . . 6 ⊢ ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11		simplll 774	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑀 ∈ (Met‘𝑋))
12		ssbnd.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
13	12	dmeqi 5737	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14		dmres 5840	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
15	13, 14	eqtri 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16		xmetf 22936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ^*)
17	16	fdmd 6497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
18	15, 17	syl5eqr 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19		df-ss 3898	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
20	18, 19	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (𝑌 × 𝑌) ⊆ dom 𝑀)
21	20	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
22		metf 22937	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
23	22	fdmd 6497	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
24	23	ad3antrrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → dom 𝑀 = (𝑋 × 𝑋))
25	21, 24	sseqtrd 3955	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26		dmss 5735	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
28		dmxpid 5764	. . . . . . . . . . . . . 14 ⊢ dom (𝑌 × 𝑌) = 𝑌
29		dmxpid 5764	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 × 𝑋) = 𝑋
30	27, 28, 29	3sstr3g 3959	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑌 ⊆ 𝑋)
31		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ 𝑌)
32	30, 31	sseldd 3916	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ 𝑋)
33		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑃 ∈ 𝑋)
34		metcl 22939	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
35	11, 32, 33, 34	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℝ)
36		rpre 12385	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
37	36	ad2antll 728	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
38	35, 37	readdcld 10659	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39		metxmet 22941	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
40	11, 39	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑀 ∈ (∞Met‘𝑋))
41	32, 31	elind 4121	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ (𝑋 ∩ 𝑌))
42		rpxr 12386	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
43	42	ad2antll 728	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
44	12	blres 23038	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
45	40, 41, 43, 44	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46		inss1 4155	. . . . . . . . . . . 12 ⊢ ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
47	35	leidd 11195	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
48	35	recnd 10658	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℂ)
49	37	recnd 10658	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℂ)
50	48, 49	pncand 10987	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
51	47, 50	breqtrrd 5058	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52		blss2 23011	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) ∧ (𝑟 ∈ ℝ ∧ ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
53	40, 32, 33, 37, 38, 51, 52	syl33anc 1382	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
54	46, 53	sstrid 3926	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
55	45, 54	eqsstrd 3953	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
57	56	sseq2d 3947	. . . . . . . . . . 11 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
58	57	rspcev 3571	. . . . . . . . . 10 ⊢ ((((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
59	38, 55, 58	syl2anc 587	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
60		sseq1 3940	. . . . . . . . . 10 ⊢ (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
61	60	rexbidv 3256	. . . . . . . . 9 ⊢ (𝑌 = (𝑦(ball‘𝑁)𝑟) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
62	59, 61	syl5ibrcom 250	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
63	62	rexlimdvva 3253	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (∃𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
64	63	expimpd 457	. . . . . 6 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
65	10, 64	syl5bi 245	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
66	65	expdimp 456	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
67	9, 66	pm2.61dne 3073	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
68	67	ex 416	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
69		simprr 772	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
70		xpss12 5534	. . . . . . 7 ⊢ ((𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
71	69, 69, 70	syl2anc 587	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
72	71	resabs1d 5849	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) = (𝑀 ↾ (𝑌 × 𝑌)))
73	72, 12	eqtr4di 2851	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) = 𝑁)
74		blbnd 35225	. . . . . . . 8 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
75	39, 74	syl3an1 1160	. . . . . . 7 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
76	75	3expa 1115	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
77	76	adantrr 716	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
78		bndss 35224	. . . . 5 ⊢ (((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) ∈ (Bnd‘𝑌))
79	77, 69, 78	syl2anc 587	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → ((𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ↾ (𝑌 × 𝑌)) ∈ (Bnd‘𝑌))
80	73, 79	eqeltrrd 2891	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
81	80	rexlimdvaa 3244	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
82	68, 81	impbid 215	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ (Bnd‘𝑌) ↔ ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 × cxp 5517 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 + caddc 10529 ℝ^*cxr 10663 ≤ cle 10665 − cmin 10859 ℝ⁺crp 12377 ∞Metcxmet 20076 Metcmet 20077 ballcbl 20078 Bndcbnd 35205

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-ec 8274 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 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 11688 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-bnd 35217

This theorem is referenced by: prdsbnd2 35233 cntotbnd 35234

W3C validator