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Theorem bndss 35224

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 22970	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
2	1	adantlr 714	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3		ssel2 3910	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
4	3	ancoms 462	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) → 𝑥 ∈ 𝑋)
5		oveq1 7142	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
6	5	eqeq2d 2809	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
7	6	rexbidv 3256	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
8	7	rspcva 3569	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
9	4, 8	sylan 583	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
10	9	adantlll 717	. . . . . . . . . 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 3899	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 = (𝑆 ∩ 𝑋))
12	11	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑆 ∩ 𝑋))
13		incom 4128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∩ 𝑋) = (𝑋 ∩ 𝑆)
14	12, 13	eqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑋 ∩ 𝑆))
15		ineq1 4131	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋 ∩ 𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
16	14, 15	sylan9eq 2853	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
17	16	adantll 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
18	17	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19		eqid 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
20	19	blssp 35194	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑆 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
21	20	an4s 659	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
22	21	anassrs 471	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
23	22	adantr 484	. . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 3233	. . . . . . . . . . 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 410	. . . . . . . . . 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 594	. . . . . . . . 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 651	. . . . . . . 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 416	. . . . . . 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 651	. . . . . 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 410	. . . . 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 651	. . . 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 3149	. . 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 515	. 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 35218	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
37	36	anbi1i 626	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋))
38		isbnd 35218	. 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 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℝ⁺crp 12377 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-mulcl 10588 ax-i2m1 10594

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-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-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-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-rp 12378 df-xadd 12496 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-bnd 35217

This theorem is referenced by: ssbnd 35226

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