ssbnd - Mathbox for Jeff Madsen

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Theorem ssbnd 37261

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 11247	. . . . . . 7 ⊢ 0 ∈ ℝ
2	1	ne0ii 4338	. . . . . 6 ⊢ ℝ ≠ ∅
3		0ss 4397	. . . . . . . 8 ⊢ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4		sseq1 4005	. . . . . . . 8 ⊢ (𝑌 = ∅ → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)))
5	3, 4	mpbiri 258	. . . . . . 7 ⊢ (𝑌 = ∅ → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
6	5	ralrimivw 3147	. . . . . 6 ⊢ (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7		r19.2z 4495	. . . . . 6 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
8	2, 6, 7	sylancr 586	. . . . 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 37256	. . . . . 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 5907	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14		dmres 6007	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
15	13, 14	eqtri 2756	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16		xmetf 24248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ^*)
17	16	fdmd 6733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
18	15, 17	eqtr3id 2782	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19		df-ss 3964	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑌 × 𝑌) ⊆ dom 𝑀 ↔ ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
20	18, 19	sylibr 233	. . . . . . . . . . . . . . . . 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 24249	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
23	22	fdmd 6733	. . . . . . . . . . . . . . . . 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 4020	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26		dmss 5905	. . . . . . . . . . . . . . 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 5932	. . . . . . . . . . . . . 14 ⊢ dom (𝑌 × 𝑌) = 𝑌
29		dmxpid 5932	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 × 𝑋) = 𝑋
30	27, 28, 29	3sstr3g 4024	. . . . . . . . . . . . 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 3981	. . . . . . . . . . . 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 24251	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
35	11, 32, 33, 34	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℝ)
36		rpre 13015	. . . . . . . . . . . 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 11274	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39		metxmet 24253	. . . . . . . . . . . . 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 4194	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ (𝑋 ∩ 𝑌))
42		rpxr 13016	. . . . . . . . . . . . 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 24350	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
45	40, 41, 43, 44	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46		inss1 4229	. . . . . . . . . . . 12 ⊢ ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
47	35	leidd 11811	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
48	35	recnd 11273	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℂ)
49	37	recnd 11273	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℂ)
50	48, 49	pncand 11603	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
51	47, 50	breqtrrd 5176	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52		blss2 24323	. . . . . . . . . . . . 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 1383	. . . . . . . . . . . 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 3991	. . . . . . . . . . 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 4018	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56		oveq2 7428	. . . . . . . . . . . 12 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
57	56	sseq2d 4012	. . . . . . . . . . 11 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
58	57	rspcev 3609	. . . . . . . . . 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 583	. . . . . . . . 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 4005	. . . . . . . . . 10 ⊢ (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
61	60	rexbidv 3175	. . . . . . . . 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 246	. . . . . . . 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 3208	. . . . . . 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 453	. . . . . 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	biimtrid 241	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
66	65	expdimp 452	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
67	9, 66	pm2.61dne 3025	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
68	67	ex 412	. 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 5693	. . . . . . 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 583	. . . . . 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 6016	. . . . 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 2786	. . . 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 37260	. . . . . . . 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 1161	. . . . . . 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 1116	. . . . . 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 37259	. . . . 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 583	. . . 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 2830	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
81	80	rexlimdvaa 3153	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) → 𝑁 ∈ (Bnd‘𝑌)))
82	68, 81	impbid 211	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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 class class class wbr 5148 × cxp 5676 dom cdm 5678 ↾ cres 5680 ‘cfv 6548 (class class class)co 7420 ℝcr 11138 0cc0 11139 + caddc 11142 ℝ^*cxr 11278 ≤ cle 11280 − cmin 11475 ℝ⁺crp 13007 ∞Metcxmet 21264 Metcmet 21265 ballcbl 21266 Bndcbnd 37240

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-ec 8727 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-bnd 37252

This theorem is referenced by: prdsbnd2 37268 cntotbnd 37269

W3C validator