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Theorem bndss 37780

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 24251	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
2	1	adantlr 715	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3		ssel2 3941	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
4	3	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) → 𝑥 ∈ 𝑋)
5		oveq1 7394	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
6	5	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
7	6	rexbidv 3157	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
8	7	rspcva 3586	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋 ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
9	4, 8	sylan 580	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟))
10	9	adantlll 718	. . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 = (𝑆 ∩ 𝑋))
12	11	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑆 ∩ 𝑋))
13		incom 4172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∩ 𝑋) = (𝑋 ∩ 𝑆)
14	12, 13	eqtrdi 2780	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑋 ∩ 𝑆))
15		ineq1 4176	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋 ∩ 𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
16	14, 15	sylan9eq 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
17	16	adantll 714	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
18	17	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
20	19	blssp 37750	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝑆 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
21	20	an4s 660	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ (𝑆 ⊆ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
22	21	anassrs 467	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥 ∈ 𝑆) ∧ 𝑆 ⊆ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
23	22	adantr 480	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3146	. . . . . . . . . . 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 406	. . . . . . . . . 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 591	. . . . . . . . 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 652	. . . . . . . 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 412	. . . . . . 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 652	. . . . . 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 406	. . . . 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 652	. . . 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 3125	. . 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 511	. 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 37774	. . 3 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
37	36	anbi1i 624	. 2 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆 ⊆ 𝑋))
38		isbnd 37774	. 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 292	1 ⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3913 ⊆ wss 3914 × cxp 5636 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 ℝ⁺crp 12951 Metcmet 21250 ballcbl 21251 Bndcbnd 37761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-mulcl 11130 ax-i2m1 11136

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-rp 12952 df-xadd 13073 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-bnd 37773

This theorem is referenced by: ssbnd 37782

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