ssbnd - Mathbox for Jeff Madsen

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Theorem ssbnd 33587

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 10040	. . . . . . 7 ⊢ 0 ∈ ℝ
2	1	ne0ii 3923	. . . . . 6 ⊢ ℝ ≠ ∅
3		0ss 3972	. . . . . . . 8 ⊢ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4		sseq1 3626	. . . . . . . 8 ⊢ (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
5	3, 4	mpbiri 248	. . . . . . 7 ⊢ (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6	5	ralrimivw 2967	. . . . . 6 ⊢ (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7		r19.2z 4060	. . . . . 6 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
8	2, 6, 7	sylancr 695	. . . . 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 33582	. . . . . 6 ⊢ ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11		simplll 798	. . . . . . . . . . . 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 5325	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14		dmres 5419	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
15	13, 14	eqtri 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16		xmetf 22134	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ^*)
17		fdm 6051	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁:(𝑌 × 𝑌)⟶ℝ^* → dom 𝑁 = (𝑌 × 𝑌))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
19	15, 18	syl5eqr 2670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
20		df-ss 3588	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
21	19, 20	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (∞Met‘𝑌) → (𝑌 × 𝑌) ⊆ dom 𝑀)
22	21	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
23		metf 22135	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
24		fdm 6051	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀:(𝑋 × 𝑋)⟶ℝ → dom 𝑀 = (𝑋 × 𝑋))
25	23, 24	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
26	25	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → dom 𝑀 = (𝑋 × 𝑋))
27	22, 26	sseqtrd 3641	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
28		dmss 5323	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
29	27, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
30		dmxpid 5345	. . . . . . . . . . . . . 14 ⊢ dom (𝑌 × 𝑌) = 𝑌
31		dmxpid 5345	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 × 𝑋) = 𝑋
32	29, 30, 31	3sstr3g 3645	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑌 ⊆ 𝑋)
33		simprl 794	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ 𝑌)
34	32, 33	sseldd 3604	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ 𝑋)
35		simpllr 799	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑃 ∈ 𝑋)
36		metcl 22137	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
37	11, 34, 35, 36	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℝ)
38		rpre 11839	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
39	38	ad2antll 765	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
40	37, 39	readdcld 10069	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
41		metxmet 22139	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
42	11, 41	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑀 ∈ (∞Met‘𝑋))
43	34, 33	elind 3798	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ (𝑋 ∩ 𝑌))
44		rpxr 11840	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
45	44	ad2antll 765	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
46	12	blres 22236	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
47	42, 43, 45, 46	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
48		inss1 3833	. . . . . . . . . . . 12 ⊢ ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
49	37	leidd 10594	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
50	37	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℂ)
51	39	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℂ)
52	50, 51	pncand 10393	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
53	49, 52	breqtrrd 4681	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
54		blss2 22209	. . . . . . . . . . . . 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)))
55	42, 34, 35, 39, 40, 53, 54	syl33anc 1341	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑀)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56	48, 55	syl5ss 3614	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
57	47, 56	eqsstrd 3639	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
58		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
59	58	sseq2d 3633	. . . . . . . . . . 11 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
60	59	rspcev 3309	. . . . . . . . . 10 ⊢ ((((𝑦𝑀𝑃) + 𝑟) ∈ ℝ ∧ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
61	40, 57, 60	syl2anc 693	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑))
62		sseq1 3626	. . . . . . . . . 10 ⊢ (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
63	62	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑌 = (𝑦(ball‘𝑁)𝑟) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∃𝑑 ∈ ℝ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
64	61, 63	syl5ibrcom 237	. . . . . . . 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)))
65	64	rexlimdvva 3038	. . . . . . 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)))
66	65	expimpd 629	. . . . . 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)))
67	10, 66	syl5bi 232	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
68	67	expdimp 453	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
69	9, 68	pm2.61dne 2880	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
70	69	ex 450	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
71		simprr 796	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
72		xpss12 5225	. . . . . . 7 ⊢ ((𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
73	71, 71, 72	syl2anc 693	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → (𝑌 × 𝑌) ⊆ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑)))
74	73	resabs1d 5428	. . . . 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)))
75	74, 12	syl6eqr 2674	. . . 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)
76		blbnd 33586	. . . . . . . 8 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
77	41, 76	syl3an1 1359	. . . . . . 7 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
78	77	3expa 1265	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑑 ∈ ℝ) → (𝑀 ↾ ((𝑃(ball‘𝑀)𝑑) × (𝑃(ball‘𝑀)𝑑))) ∈ (Bnd‘(𝑃(ball‘𝑀)𝑑)))
79	78	adantrr 753	. . . . 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)))
80		bndss 33585	. . . . 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))
81	79, 71, 80	syl2anc 693	. . . 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))
82	75, 81	eqeltrrd 2702	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
83	82	rexlimdvaa 3032	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
84	70, 83	impbid 202	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 × cxp 5112 dom cdm 5114 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 + caddc 9939 ℝ^*cxr 10073 ≤ cle 10075 − cmin 10266 ℝ⁺crp 11832 ∞Metcxmt 19731 Metcme 19732 ballcbl 19733 Bndcbnd 33566

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-ec 7744 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-bnd 33578

This theorem is referenced by: prdsbnd2 33594 cntotbnd 33595

W3C validator