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Theorem blval 13183
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 13180 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
213ad2ant1 1013 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
3 simprl 526 . . . . 5 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑦 = 𝑃)
43oveq1d 5868 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥))
5 simprr 527 . . . 4 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑟 = 𝑅)
64, 5breq12d 4002 . . 3 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅))
76rabbidv 2719 . 2 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
8 simp2 993 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑃𝑋)
9 simp3 994 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
10 xmetrel 13137 . . . . 5 Rel ∞Met
11 relelfvdm 5528 . . . . 5 ((Rel ∞Met ∧ 𝐷 ∈ (∞Met‘𝑋)) → 𝑋 ∈ dom ∞Met)
1210, 11mpan 422 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
13123ad2ant1 1013 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑋 ∈ dom ∞Met)
14 rabexg 4132 . . 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 5980 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 973   = wceq 1348  wcel 2141  {crab 2452  Vcvv 2730   class class class wbr 3989  dom cdm 4611  Rel wrel 4616  cfv 5198  (class class class)co 5853  cmpo 5855  *cxr 7953   < clt 7954  ∞Metcxmet 12774  ballcbl 12776
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-psmet 12781  df-xmet 12782  df-bl 12784
This theorem is referenced by:  elbl  13185  metss2lem  13291  bdbl  13297  xmetxpbl  13302
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