ssbnd - Mathbox for Jeff Madsen

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Theorem ssbnd 38925

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 11340	. . . . . . 7 ⊢ 0 ∈ ℝ
2	1	ne0ii 4293	. . . . . 6 ⊢ ℝ ≠ ∅
3		0ss 4353	. . . . . . . 8 ⊢ ∅ ⊆ (𝑃(ball‘𝑀)𝑑)
4		sseq1 3938	. . . . . . . 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 3162	. . . . . 6 ⊢ (𝑌 = ∅ → ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
7		r19.2z 4458	. . . . . 6 ⊢ ((ℝ ≠ ∅ ∧ ∀𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
8	2, 6, 7	sylancr 599	. . . . 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 38920	. . . . . 6 ⊢ ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∃𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
11		simplll 787	. . . . . . . . . . . 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 5894	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom 𝑁 = dom (𝑀 ↾ (𝑌 × 𝑌))
14		dmres 6016	. . . . . . . . . . . . . . . . . . . 20 ⊢ dom (𝑀 ↾ (𝑌 × 𝑌)) = ((𝑌 × 𝑌) ∩ dom 𝑀)
15	13, 14	eqtri 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝑁 = ((𝑌 × 𝑌) ∩ dom 𝑀)
16		xmetf 24681	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ^*)
17	16	fdmd 6745	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
18	15, 17	eqtr3id 2812	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (∞Met‘𝑌) → ((𝑌 × 𝑌) ∩ dom 𝑀) = (𝑌 × 𝑌))
19		dfss2 4097	. . . . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ dom 𝑀)
22		metf 24682	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
23	22	fdmd 6745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
24	23	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → dom 𝑀 = (𝑋 × 𝑋))
25	21, 24	sseqtrd 3949	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
26		dmss 5892	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
27	25, 26	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
28		dmxpid 5920	. . . . . . . . . . . . . 14 ⊢ dom (𝑌 × 𝑌) = 𝑌
29		dmxpid 5920	. . . . . . . . . . . . . 14 ⊢ dom (𝑋 × 𝑋) = 𝑋
30	27, 28, 29	3sstr3g 3965	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑌 ⊆ 𝑋)
31		simprl 783	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ 𝑌)
32	30, 31	sseldd 3914	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ 𝑋)
33		simpllr 788	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑃 ∈ 𝑋)
34		metcl 24684	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝑀𝑃) ∈ ℝ)
35	11, 32, 33, 34	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℝ)
36		rpre 13159	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
37	36	ad2antll 742	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ)
38	35, 37	readdcld 11368	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝑦𝑀𝑃) + 𝑟) ∈ ℝ)
39		metxmet 24686	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
40	11, 39	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑀 ∈ (∞Met‘𝑋))
41	32, 31	elind 4100	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑦 ∈ (𝑋 ∩ 𝑌))
42		rpxr 13160	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
43	42	ad2antll 742	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
44	12	blres 24783	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
45	40, 41, 43, 44	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) = ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌))
46		inss1 4136	. . . . . . . . . . . 12 ⊢ ((𝑦(ball‘𝑀)𝑟) ∩ 𝑌) ⊆ (𝑦(ball‘𝑀)𝑟)
47	35	leidd 11912	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (𝑦𝑀𝑃))
48	35	recnd 11367	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ∈ ℂ)
49	37	recnd 11367	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℂ)
50	48, 49	pncand 11700	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (((𝑦𝑀𝑃) + 𝑟) − 𝑟) = (𝑦𝑀𝑃))
51	47, 50	breqtrrd 5137	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦𝑀𝑃) ≤ (((𝑦𝑀𝑃) + 𝑟) − 𝑟))
52		blss2 24756	. . . . . . . . . . . . 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 1413	. . . . . . . . . . . 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 3924	. . . . . . . . . . 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 3947	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
56		oveq2 7454	. . . . . . . . . . . 12 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → (𝑃(ball‘𝑀)𝑑) = (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟)))
57	56	sseq2d 3945	. . . . . . . . . . 11 ⊢ (𝑑 = ((𝑦𝑀𝑃) + 𝑟) → ((𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)((𝑦𝑀𝑃) + 𝑟))))
58	57	rspcev 3580	. . . . . . . . . 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 596	. . . . . . . . 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 3938	. . . . . . . . . 10 ⊢ (𝑌 = (𝑦(ball‘𝑁)𝑟) → (𝑌 ⊆ (𝑃(ball‘𝑀)𝑑) ↔ (𝑦(ball‘𝑁)𝑟) ⊆ (𝑃(ball‘𝑀)𝑑)))
61	60	rexbidv 3190	. . . . . . . . 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 3223	. . . . . . 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 459	. . . . . 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 245	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑁 ∈ (Bnd‘𝑌) ∧ 𝑌 ≠ ∅) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
66	65	expdimp 458	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → (𝑌 ≠ ∅ → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
67	9, 66	pm2.61dne 3045	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ (Bnd‘𝑌)) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
68	67	ex 418	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ (Bnd‘𝑌) → ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
69		simprr 785	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))
70		xpss12 5671	. . . . . . 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 596	. . . . . 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 6012	. . . . 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 2816	. . . 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 38924	. . . . . . . 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 1181	. . . . . . 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 1136	. . . . . 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 730	. . . . 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 38923	. . . . 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 596	. . . 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 2865	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑑 ∈ ℝ ∧ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) → 𝑁 ∈ (Bnd‘𝑌))
81	80	rexlimdvaa 3168	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 class class class wbr 5106 × cxp 5654 dom cdm 5656 ↾ cres 5658 ‘cfv 6561 (class class class)co 7446 ℝcr 11229 0cc0 11230 + caddc 11233 ℝ^*cxr 11372 ≤ cle 11374 − cmin 11571 ℝ⁺crp 13150 ∞Metcxmet 21696 Metcmet 21697 ballcbl 21698 Bndcbnd 38904

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-er 8748 df-ec 8750 df-map 8880 df-en 9005 df-dom 9006 df-sdom 9007 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-bnd 38916

This theorem is used by: prdsbnd2 38932 cntotbnd 38933

W3C validator