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Theorem bndss 34072
Description: A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
bndss ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))

Proof of Theorem bndss
Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metres2 22496 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
21adantlr 707 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3 ssel2 3793 . . . . . . . . . . . . 13 ((𝑆𝑋𝑥𝑆) → 𝑥𝑋)
43ancoms 451 . . . . . . . . . . . 12 ((𝑥𝑆𝑆𝑋) → 𝑥𝑋)
5 oveq1 6885 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
65eqeq2d 2809 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
76rexbidv 3233 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
87rspcva 3495 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
94, 8sylan 576 . . . . . . . . . . 11 (((𝑥𝑆𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
109adantlll 710 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
11 dfss 3784 . . . . . . . . . . . . . . . . . . 19 (𝑆𝑋𝑆 = (𝑆𝑋))
1211biimpi 208 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋𝑆 = (𝑆𝑋))
13 incom 4003 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋) = (𝑋𝑆)
1412, 13syl6eq 2849 . . . . . . . . . . . . . . . . 17 (𝑆𝑋𝑆 = (𝑋𝑆))
15 ineq1 4005 . . . . . . . . . . . . . . . . 17 (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1614, 15sylan9eq 2853 . . . . . . . . . . . . . . . 16 ((𝑆𝑋𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1716adantll 706 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1817adantlr 707 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19 eqid 2799 . . . . . . . . . . . . . . . . . 18 (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
2019blssp 34039 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝑥𝑆𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2120an4s 651 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ (𝑆𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2221anassrs 460 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2322adantr 473 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2418, 23eqtr4d 2836 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2524ex 402 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2625reximdva 3197 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2726imp 396 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2810, 27syldan 586 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2928an32s 643 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3029ex 402 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3130an32s 643 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3231imp 396 . . . . 5 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3332an32s 643 . . . 4 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) ∧ 𝑥𝑆) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3433ralrimiva 3147 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
352, 34jca 508 . 2 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
36 isbnd 34066 . . 3 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
3736anbi1i 618 . 2 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋))
38 isbnd 34066 . 2 ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆) ↔ ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3935, 37, 383imtr4i 284 1 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  cin 3768  wss 3769   × cxp 5310  cres 5314  cfv 6101  (class class class)co 6878  +crp 12074  Metcmet 20054  ballcbl 20055  Bndcbnd 34053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-mulcl 10286  ax-i2m1 10292
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-rp 12075  df-xadd 12194  df-psmet 20060  df-xmet 20061  df-met 20062  df-bl 20063  df-bnd 34065
This theorem is referenced by:  ssbnd  34074
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