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Theorem blvalps 13521
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blvalps ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Distinct variable groups:   𝑥,𝑃   𝑥,𝐷   𝑥,𝑅   𝑥,𝑋

Proof of Theorem blvalps
Dummy variables 𝑟 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfvalps 13518 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
213ad2ant1 1018 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦𝑋, 𝑟 ∈ ℝ* ↦ {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
3 simprl 529 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑦 = 𝑃)
43oveq1d 5883 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥))
5 simprr 531 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → 𝑟 = 𝑅)
64, 5breq12d 4013 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅))
76rabbidv 2726 . 2 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃𝑟 = 𝑅)) → {𝑥𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
8 simp2 998 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑃𝑋)
9 simp3 999 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
10 psmetrel 13455 . . . . 5 Rel PsMet
11 relelfvdm 5542 . . . . 5 ((Rel PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ dom PsMet)
1210, 11mpan 424 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
13123ad2ant1 1018 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → 𝑋 ∈ dom PsMet)
14 rabexg 4143 . . 3 (𝑋 ∈ dom PsMet → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
1513, 14syl 14 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
162, 7, 8, 9, 15ovmpod 5995 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  {crab 2459  Vcvv 2737   class class class wbr 4000  dom cdm 4622  Rel wrel 4627  cfv 5211  (class class class)co 5868  cmpo 5870  *cxr 7968   < clt 7969  PsMetcpsmet 13112  ballcbl 13115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-pnf 7971  df-mnf 7972  df-xr 7973  df-psmet 13120  df-bl 13123
This theorem is referenced by:  elblps  13523
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