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Theorem bndss 38418

Description: A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Assertion**
Ref	Expression
bndss	⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))

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

Step	Hyp	Ref	Expression
1		metres2 24501	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
2	1	adantlr 727	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3		ssel2 3933	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
4	3	ancoms 463	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) → 𝑥 ∈ 𝑋)
5		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
6	5	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
7	6	rexbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
8	7	rspcva 3580	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
9	4, 8	sylan 591	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
10	9	adantlll 730	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
11		dfss 3925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 = (𝑆 ∩ 𝑋))
12	11	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑆 ∩ 𝑋))
13		incom 4163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∩ 𝑋) = (𝑋 ∩ 𝑆)
14	12, 13	eqtrdi 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑋 ∩ 𝑆))
15		ineq1 4167	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋 ∩ 𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
16	14, 15	sylan9eq 2818	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
17	16	adantll 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
18	17	adantlr 727	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
20	19	blssp 38388	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑆 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
21	20	an4s 672	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
22	21	anassrs 472	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
23	22	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
24	18, 23	eqtr4d 2801	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
25	24	ex 417	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
26	25	reximdva 3178	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
27	26	imp 411	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
28	10, 27	syldan 602	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
29	28	an32s 664	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
30	29	ex 417	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → (𝑆 ⊆ 𝑋 → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
31	30	an32s 664	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥 ∈ 𝑆) → (𝑆 ⊆ 𝑋 → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
32	31	imp 411	. . . . 5 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
33	32	an32s 664	. . . 4 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ 𝑆) → ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
34	33	ralrimiva 3157	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → ∀𝑥 ∈ 𝑆 ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
35	2, 34	jca 520	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥 ∈ 𝑆 ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
36		isbnd 38412	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
37	36	anbi1i 635	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋))
38		isbnd 38412	. 2 ⊢ ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆) ↔ ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥 ∈ 𝑆 ∃𝑟 ∈ ℝ⁺ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
39	35, 37, 38	3imtr4i 295	1 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∩ cin 3905 ⊆ wss 3906 × cxp 5661 ↾ cres 5665 ‘cfv 6538 (class class class)co 7412 ℝ⁺crp 13017 Metcmet 21489 ballcbl 21490 Bndcbnd 38399

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-mulcl 11163 ax-i2m1 11169

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-rp 13018 df-xadd 13139 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-bnd 38411

This theorem is referenced by: ssbnd 38420

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