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Theorem blval 24412
Description: The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Assertion
Ref Expression
blval ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Distinct variable groups:   𝑥,𝑃   𝑥,𝐷   𝑥,𝑅   𝑥,𝑋

Proof of Theorem blval
Dummy variables 𝑟 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfval 24410 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
213ad2ant1 1132 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
3 simprl 771 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑦 = 𝑃)
43oveq1d 7446 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥))
5 simprr 773 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑟 = 𝑅)
64, 5breq12d 5161 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅))
76rabbidv 3441 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
8 simp2 1136 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑃𝑋)
9 simp3 1137 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
10 elfvdm 6944 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
11103ad2ant1 1132 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑋 ∈ dom ∞Met)
12 rabexg 5343 . . 3 (𝑋 ∈ dom ∞Met → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
1311, 12syl 17 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
142, 7, 8, 9, 13ovmpod 7585 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478   class class class wbr 5148  dom cdm 5689  cfv 6563  (class class class)co 7431  cmpo 7433  *cxr 11292   < clt 11293  ∞Metcxmet 21367  ballcbl 21369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-xr 11297  df-psmet 21374  df-xmet 21375  df-bl 21377
This theorem is referenced by:  elbl  24414  metss2lem  24540  stdbdbl  24546  nmhmcn  25167  lgamucov  27096  isbnd3  37771
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