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Theorem bndss 33214
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 22078 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
21adantlr 750 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆))
3 ssel2 3578 . . . . . . . . . . . . 13 ((𝑆𝑋𝑥𝑆) → 𝑥𝑋)
43ancoms 469 . . . . . . . . . . . 12 ((𝑥𝑆𝑆𝑋) → 𝑥𝑋)
5 oveq1 6611 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑟) = (𝑥(ball‘𝑀)𝑟))
65eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
76rexbidv 3045 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
87rspcva 3293 . . . . . . . . . . . 12 ((𝑥𝑋 ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
94, 8sylan 488 . . . . . . . . . . 11 (((𝑥𝑆𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
109adantlll 753 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))
11 dfss 3570 . . . . . . . . . . . . . . . . . . 19 (𝑆𝑋𝑆 = (𝑆𝑋))
1211biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋𝑆 = (𝑆𝑋))
13 incom 3783 . . . . . . . . . . . . . . . . . 18 (𝑆𝑋) = (𝑋𝑆)
1412, 13syl6eq 2671 . . . . . . . . . . . . . . . . 17 (𝑆𝑋𝑆 = (𝑋𝑆))
15 ineq1 3785 . . . . . . . . . . . . . . . . 17 (𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑋𝑆) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1614, 15sylan9eq 2675 . . . . . . . . . . . . . . . 16 ((𝑆𝑋𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1716adantll 749 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
1817adantlr 750 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
19 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑀 ↾ (𝑆 × 𝑆)) = (𝑀 ↾ (𝑆 × 𝑆))
2019blssp 33181 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝑥𝑆𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2120an4s 868 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ (𝑆𝑋𝑟 ∈ ℝ+)) → (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟) = ((𝑥(ball‘𝑀)𝑟) ∩ 𝑆))
2221anassrs 679 . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2524ex 450 . . . . . . . . . . . 12 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑋 = (𝑥(ball‘𝑀)𝑟) → 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2625reximdva 3011 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
2726imp 445 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2810, 27syldan 487 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ 𝑆𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
2928an32s 845 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3029ex 450 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑥𝑆) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3130an32s 845 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) → (𝑆𝑋 → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟)))
3231imp 445 . . . . 5 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑥𝑆) ∧ 𝑆𝑋) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3332an32s 845 . . . 4 ((((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) ∧ 𝑥𝑆) → ∃𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
3433ralrimiva 2960 . . 3 (((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋) → ∀𝑥𝑆𝑟 ∈ ℝ+ 𝑆 = (𝑥(ball‘(𝑀 ↾ (𝑆 × 𝑆)))𝑟))
352, 34jca 554 . 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 33208 . . 3 (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)))
3736anbi1i 730 . 2 ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) ↔ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑟)) ∧ 𝑆𝑋))
38 isbnd 33208 . 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555   × cxp 5072  cres 5076  cfv 5847  (class class class)co 6604  +crp 11776  Metcme 19651  ballcbl 19652  Bndcbnd 33195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-mulcl 9942  ax-i2m1 9948
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-rp 11777  df-xadd 11891  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-bnd 33207
This theorem is referenced by:  ssbnd  33216
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