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Theorem bndss 34933
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 22888 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
21adantlr 711 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3 ssel2 3965 . . . . . . . . . . . . 13 ((𝑆𝑋𝑥𝑆) → 𝑥𝑋)
43ancoms 459 . . . . . . . . . . . 12 ((𝑥𝑆𝑆𝑋) → 𝑥𝑋)
5 oveq1 7158 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
65eqeq2d 2835 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
76rexbidv 3301 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
87rspcva 3624 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
94, 8sylan 580 . . . . . . . . . . 11 (((𝑥𝑆𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
109adantlll 714 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
11 dfss 3956 . . . . . . . . . . . . . . . . . . 19 (𝑆𝑋𝑆 = (𝑆𝑋))
1211biimpi 217 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋𝑆 = (𝑆𝑋))
13 incom 4181 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋) = (𝑋𝑆)
1412, 13syl6eq 2876 . . . . . . . . . . . . . . . . 17 (𝑆𝑋𝑆 = (𝑋𝑆))
15 ineq1 4184 . . . . . . . . . . . . . . . . 17 (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1614, 15sylan9eq 2880 . . . . . . . . . . . . . . . 16 ((𝑆𝑋𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1716adantll 710 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1817adantlr 711 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19 eqid 2824 . . . . . . . . . . . . . . . . . 18 (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
2019blssp 34900 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝑥𝑆𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2120an4s 656 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ (𝑆𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2221anassrs 468 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2322adantr 481 . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2524ex 413 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2625reximdva 3278 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2726imp 407 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2810, 27syldan 591 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2928an32s 648 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3029ex 413 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3130an32s 648 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3231imp 407 . . . . 5 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3332an32s 648 . . . 4 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) ∧ 𝑥𝑆) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3433ralrimiva 3186 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
352, 34jca 512 . 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 34927 . . 3 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
3736anbi1i 623 . 2 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋))
38 isbnd 34927 . 2 ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆) ↔ ((𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ∧ ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3935, 37, 383imtr4i 293 1 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  wral 3142  wrex 3143  cin 3938  wss 3939   × cxp 5551  cres 5555  cfv 6351  (class class class)co 7151  +crp 12382  Metcmet 20447  ballcbl 20448  Bndcbnd 34914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-mulcl 10591  ax-i2m1 10597
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-rp 12383  df-xadd 12501  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-bnd 34926
This theorem is referenced by:  ssbnd  34935
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