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Theorem spheres 45090
 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 45074 . . 3 Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4 fveq2 6661 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 spheres.b . . . . . . 7 𝐵 = (Base‘𝑊)
65eqcomi 2833 . . . . . 6 (Base‘𝑊) = 𝐵
76a1i 11 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
84, 7eqtrd 2859 . . . 4 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9 eqidd 2825 . . . 4 (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10 fveq2 6661 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11 spheres.d . . . . . . . . . 10 𝐷 = (dist‘𝑊)
1211eqcomi 2833 . . . . . . . . 9 (dist‘𝑊) = 𝐷
1312a1i 11 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
1410, 13eqtrd 2859 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
1514oveqd 7166 . . . . . 6 (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
1615eqeq1d 2826 . . . . 5 (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
178, 16rabeqbidv 3471 . . . 4 (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
188, 9, 17mpoeq123dv 7222 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}) = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
19 elex 3498 . . 3 (𝑊𝑉𝑊 ∈ V)
20 fvex 6674 . . . . . 6 (Base‘𝑊) ∈ V
215, 20eqeltri 2912 . . . . 5 𝐵 ∈ V
22 ovex 7182 . . . . 5 (0[,]+∞) ∈ V
2321, 22mpoex 7773 . . . 4 (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V
2423a1i 11 . . 3 (𝑊𝑉 → (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V)
253, 18, 19, 24fvmptd3 6782 . 2 (𝑊𝑉 → (Sphere‘𝑊) = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
262, 25eqtrd 2859 1 (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  {crab 3137  Vcvv 3480  ‘cfv 6343  (class class class)co 7149   ∈ cmpo 7151  0cc0 10535  +∞cpnf 10670  [,]cicc 12738  Basecbs 16483  distcds 16574  Spherecsph 45072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-sph 45074 This theorem is referenced by:  sphere  45091  rrxsphere  45092
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