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Theorem blval 12789
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 12786 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
213ad2ant1 1003 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
3 simprl 521 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑦 = 𝑃)
43oveq1d 5839 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥))
5 simprr 522 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑟 = 𝑅)
64, 5breq12d 3978 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅))
76rabbidv 2701 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
8 simp2 983 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑃𝑋)
9 simp3 984 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
10 xmetrel 12743 . . . . 5 Rel ∞Met
11 relelfvdm 5500 . . . . 5 ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met)
1210, 11mpan 421 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
13123ad2ant1 1003 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑋 ∈ dom ∞Met)
14 rabexg 4107 . . 3 (𝑋 ∈ dom ∞Met → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
1513, 14syl 14 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
162, 7, 8, 9, 15ovmpod 5948 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1335  wcel 2128  {crab 2439  Vcvv 2712   class class class wbr 3965  dom cdm 4586  Rel wrel 4591  cfv 5170  (class class class)co 5824  cmpo 5826  *cxr 7911   < clt 7912  ∞Metcxmet 12380  ballcbl 12382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-map 6595  df-pnf 7914  df-mnf 7915  df-xr 7916  df-psmet 12387  df-xmet 12388  df-bl 12390
This theorem is referenced by:  elbl  12791  metss2lem  12897  bdbl  12903  xmetxpbl  12908
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