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Theorem spheres 48667
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 48651 . . 3 Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4 fveq2 6906 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 spheres.b . . . . . . 7 𝐵 = (Base‘𝑊)
65eqcomi 2746 . . . . . 6 (Base‘𝑊) = 𝐵
76a1i 11 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
84, 7eqtrd 2777 . . . 4 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9 eqidd 2738 . . . 4 (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10 fveq2 6906 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11 spheres.d . . . . . . . . . 10 𝐷 = (dist‘𝑊)
1211eqcomi 2746 . . . . . . . . 9 (dist‘𝑊) = 𝐷
1312a1i 11 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
1410, 13eqtrd 2777 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
1514oveqd 7448 . . . . . 6 (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
1615eqeq1d 2739 . . . . 5 (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
178, 16rabeqbidv 3455 . . . 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 3501 . . 3 (𝑊𝑉𝑊 ∈ V)
20 fvex 6919 . . . . . 6 (Base‘𝑊) ∈ V
215, 20eqeltri 2837 . . . . 5 𝐵 ∈ V
22 ovex 7464 . . . . 5 (0[,]+∞) ∈ V
2321, 22mpoex 8104 . . . 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 2777 1 (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  Basecbs 17247  distcds 17306  Spherecsph 48649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-sph 48651
This theorem is referenced by:  sphere  48668  rrxsphere  48669
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