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Theorem bndss 38245

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 24410	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
2	1	adantlr 725	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3		ssel2 3929	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
4	3	ancoms 462	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) → 𝑥 ∈ 𝑋)
5		oveq1 7397	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
6	5	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
7	6	rexbidv 3185	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
8	7	rspcva 3578	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
9	4, 8	sylan 589	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
10	9	adantlll 728	. . . . . . . . . 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 3921	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 = (𝑆 ∩ 𝑋))
12	11	biimpi 218	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑆 ∩ 𝑋))
13		incom 4159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∩ 𝑋) = (𝑋 ∩ 𝑆)
14	12, 13	eqtrdi 2812	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑋 ∩ 𝑆))
15		ineq1 4163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋 ∩ 𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
16	14, 15	sylan9eq 2816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
17	16	adantll 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
18	17	adantlr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19		eqid 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
20	19	blssp 38215	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑆 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
21	20	an4s 670	. . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . 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 3174	. . . . . . . . . . 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 600	. . . . . . . . 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 662	. . . . . . . 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 662	. . . . . 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 662	. . . 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 3153	. . 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 519	. 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 38239	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
37	36	anbi1i 633	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋))
38		isbnd 38239	. 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 294	1 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ∩ cin 3901 ⊆ wss 3902 × cxp 5641 ↾ cres 5645 ‘cfv 6515 (class class class)co 7390 ℝ⁺crp 12986 Metcmet 21397 ballcbl 21398 Bndcbnd 38226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-mulcl 11128 ax-i2m1 11134

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7964 df-2nd 7965 df-er 8671 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-rp 12987 df-xadd 13108 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-bnd 38238

This theorem is referenced by: ssbnd 38247

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