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Theorem bndss 37787

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 24258	. . . 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 3944	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑋)
4	3	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑆 ⊆ 𝑋) → 𝑥 ∈ 𝑋)
5		oveq1 7397	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
6	5	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
7	6	rexbidv 3158	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
8	7	rspcva 3589	. . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 = (𝑆 ∩ 𝑋))
12	11	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑆 ∩ 𝑋))
13		incom 4175	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∩ 𝑋) = (𝑋 ∩ 𝑆)
14	12, 13	eqtrdi 2781	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ 𝑋 → 𝑆 = (𝑋 ∩ 𝑆))
15		ineq1 4179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋 ∩ 𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
16	14, 15	sylan9eq 2785	. . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
20	19	blssp 37757	. . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 3147	. . . . . . . . . . 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 3126	. . 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 37781	. . 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 37781	. 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 3045 ∃wrex 3054 ∩ cin 3916 ⊆ wss 3917 × cxp 5639 ↾ cres 5643 ‘cfv 6514 (class class class)co 7390 ℝ⁺crp 12958 Metcmet 21257 ballcbl 21258 Bndcbnd 37768

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-mulcl 11137 ax-i2m1 11143

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-rp 12959 df-xadd 13080 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-bnd 37780

This theorem is referenced by: ssbnd 37789

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