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Theorem spheres 43492
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 43476 . . 3 Sphere = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}))
4 fveq2 6448 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 spheres.b . . . . . . 7 𝐵 = (Base‘𝑊)
65eqcomi 2787 . . . . . 6 (Base‘𝑊) = 𝐵
76a1i 11 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑊) = 𝐵)
84, 7eqtrd 2814 . . . 4 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
9 eqidd 2779 . . . 4 (𝑤 = 𝑊 → (0[,]+∞) = (0[,]+∞))
10 fveq2 6448 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑤) = (dist‘𝑊))
11 spheres.d . . . . . . . . . 10 𝐷 = (dist‘𝑊)
1211eqcomi 2787 . . . . . . . . 9 (dist‘𝑊) = 𝐷
1312a1i 11 . . . . . . . 8 (𝑤 = 𝑊 → (dist‘𝑊) = 𝐷)
1410, 13eqtrd 2814 . . . . . . 7 (𝑤 = 𝑊 → (dist‘𝑤) = 𝐷)
1514oveqd 6941 . . . . . 6 (𝑤 = 𝑊 → (𝑝(dist‘𝑤)𝑥) = (𝑝𝐷𝑥))
1615eqeq1d 2780 . . . . 5 (𝑤 = 𝑊 → ((𝑝(dist‘𝑤)𝑥) = 𝑟 ↔ (𝑝𝐷𝑥) = 𝑟))
178, 16rabeqbidv 3392 . . . 4 (𝑤 = 𝑊 → {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟} = {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟})
188, 9, 17mpt2eq123dv 6996 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤), 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ (Base‘𝑤) ∣ (𝑝(dist‘𝑤)𝑥) = 𝑟}) = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
19 elex 3414 . . 3 (𝑊𝑉𝑊 ∈ V)
20 fvex 6461 . . . . . 6 (Base‘𝑊) ∈ V
215, 20eqeltri 2855 . . . . 5 𝐵 ∈ V
22 ovex 6956 . . . . 5 (0[,]+∞) ∈ V
2321, 22mpt2ex 7529 . . . 4 (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V
2423a1i 11 . . 3 (𝑊𝑉 → (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}) ∈ V)
253, 18, 19, 24fvmptd3 6566 . 2 (𝑊𝑉 → (Sphere‘𝑊) = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
262, 25eqtrd 2814 1 (𝑊𝑉𝑆 = (𝑥𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  cfv 6137  (class class class)co 6924  cmpt2 6926  0cc0 10274  +∞cpnf 10410  [,]cicc 12494  Basecbs 16259  distcds 16351  Spherecsph 43474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-sph 43476
This theorem is referenced by:  sphere  43493  rrxsphere  43494
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