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Theorem spheres 48596
Description: The spheres for given centers and radii in a metric space (or any extensible structure having a base set and a distance function). (Contributed by AV, 22-Jan-2023.)
Hypotheses
Ref Expression
spheres.b 𝐵 = (Base‘𝑊)
spheres.l 𝑆 = (Sphere‘𝑊)
spheres.d 𝐷 = (dist‘𝑊)
Assertion
Ref Expression
spheres (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
Distinct variable groups:   𝐵,𝑝,𝑟,𝑥   𝑊,𝑝,𝑟,𝑥
Allowed substitution hints:   𝐷(𝑥,𝑟,𝑝)   𝑆(𝑥,𝑟,𝑝)   𝑉(𝑥,𝑟,𝑝)

Proof of Theorem spheres
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 spheres.l . . 3 𝑆 = (Sphere‘𝑊)
21a1i 11 . 2 (𝑊𝑉𝑆 = (Sphere‘𝑊))
3 df-sph 48580 . . 3 Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4 fveq2 6907 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 spheres.b . . . . . . 7 𝐵 = (Base‘𝑊)
65eqcomi 2744 . . . . . 6 (Base‘𝑊) = 𝐵
76a1i 11 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
84, 7eqtrd 2775 . . . 4 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9 eqidd 2736 . . . 4 (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10 fveq2 6907 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11 spheres.d . . . . . . . . . 10 𝐷 = (dist‘𝑊)
1211eqcomi 2744 . . . . . . . . 9 (dist‘𝑊) = 𝐷
1312a1i 11 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
1410, 13eqtrd 2775 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
1514oveqd 7448 . . . . . 6 (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
1615eqeq1d 2737 . . . . 5 (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
178, 16rabeqbidv 3452 . . . 4 (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
188, 9, 17mpoeq123dv 7508 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}) = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
19 elex 3499 . . 3 (𝑊𝑉𝑊 ∈ V)
20 fvex 6920 . . . . . 6 (Base‘𝑊) ∈ V
215, 20eqeltri 2835 . . . . 5 𝐵 ∈ V
22 ovex 7464 . . . . 5 (0[,]+∞) ∈ V
2321, 22mpoex 8103 . . . 4 (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V
2423a1i 11 . . 3 (𝑊𝑉 → (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V)
253, 18, 19, 24fvmptd3 7039 . 2 (𝑊𝑉 → (Sphere‘𝑊) = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
262, 25eqtrd 2775 1 (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  +∞cpnf 11290  [,]cicc 13387  Basecbs 17245  distcds 17307  Spherecsph 48578
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
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-reu 3379  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-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-sph 48580
This theorem is referenced by:  sphere  48597  rrxsphere  48598
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