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Theorem bndss 33890
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 22361 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
21adantlr 753 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3 ssel2 3731 . . . . . . . . . . . . 13 ((𝑆𝑋𝑥𝑆) → 𝑥𝑋)
43ancoms 468 . . . . . . . . . . . 12 ((𝑥𝑆𝑆𝑋) → 𝑥𝑋)
5 oveq1 6812 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
65eqeq2d 2762 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
76rexbidv 3182 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
87rspcva 3439 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
94, 8sylan 489 . . . . . . . . . . 11 (((𝑥𝑆𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
109adantlll 756 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
11 dfss 3722 . . . . . . . . . . . . . . . . . . 19 (𝑆𝑋𝑆 = (𝑆𝑋))
1211biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋𝑆 = (𝑆𝑋))
13 incom 3940 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋) = (𝑋𝑆)
1412, 13syl6eq 2802 . . . . . . . . . . . . . . . . 17 (𝑆𝑋𝑆 = (𝑋𝑆))
15 ineq1 3942 . . . . . . . . . . . . . . . . 17 (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1614, 15sylan9eq 2806 . . . . . . . . . . . . . . . 16 ((𝑆𝑋𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1716adantll 752 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1817adantlr 753 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19 eqid 2752 . . . . . . . . . . . . . . . . . 18 (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
2019blssp 33857 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝑥𝑆𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2120an4s 904 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ (𝑆𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2221anassrs 683 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2322adantr 472 . . . . . . . . . . . . . 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 2789 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2524ex 449 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2625reximdva 3147 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2726imp 444 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2810, 27syldan 488 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2928an32s 881 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3029ex 449 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3130an32s 881 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3231imp 444 . . . . 5 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3332an32s 881 . . . 4 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) ∧ 𝑥𝑆) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3433ralrimiva 3096 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
352, 34jca 555 . 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 33884 . . 3 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
3736anbi1i 733 . 2 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋))
38 isbnd 33884 . 2 ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆) ↔ ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3935, 37, 383imtr4i 281 1 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  wrex 3043  cin 3706  wss 3707   × cxp 5256  cres 5260  cfv 6041  (class class class)co 6805  +crp 12017  Metcme 19926  ballcbl 19927  Bndcbnd 33871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-mulcl 10182  ax-i2m1 10188
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-rp 12018  df-xadd 12132  df-psmet 19932  df-xmet 19933  df-met 19934  df-bl 19935  df-bnd 33883
This theorem is referenced by:  ssbnd  33892
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