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Theorem spheres 46733
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 𝐡 = (Baseβ€˜π‘Š)
spheres.l 𝑆 = (Sphereβ€˜π‘Š)
spheres.d 𝐷 = (distβ€˜π‘Š)
Assertion
Ref Expression
spheres (π‘Š ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
Distinct variable groups:   𝐡,𝑝,π‘Ÿ,π‘₯   π‘Š,𝑝,π‘Ÿ,π‘₯
Allowed substitution hints:   𝐷(π‘₯,π‘Ÿ,𝑝)   𝑆(π‘₯,π‘Ÿ,𝑝)   𝑉(π‘₯,π‘Ÿ,𝑝)

Proof of Theorem spheres
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 spheres.l . . 3 𝑆 = (Sphereβ€˜π‘Š)
21a1i 11 . 2 (π‘Š ∈ 𝑉 β†’ 𝑆 = (Sphereβ€˜π‘Š))
3 df-sph 46717 . . 3 Sphere = (𝑀 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘€), π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ (Baseβ€˜π‘€) ∣ (𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ}))
4 fveq2 6839 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
5 spheres.b . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
65eqcomi 2746 . . . . . 6 (Baseβ€˜π‘Š) = 𝐡
76a1i 11 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜π‘Š) = 𝐡)
84, 7eqtrd 2777 . . . 4 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝐡)
9 eqidd 2738 . . . 4 (𝑀 = π‘Š β†’ (0[,]+∞) = (0[,]+∞))
10 fveq2 6839 . . . . . . . 8 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = (distβ€˜π‘Š))
11 spheres.d . . . . . . . . . 10 𝐷 = (distβ€˜π‘Š)
1211eqcomi 2746 . . . . . . . . 9 (distβ€˜π‘Š) = 𝐷
1312a1i 11 . . . . . . . 8 (𝑀 = π‘Š β†’ (distβ€˜π‘Š) = 𝐷)
1410, 13eqtrd 2777 . . . . . . 7 (𝑀 = π‘Š β†’ (distβ€˜π‘€) = 𝐷)
1514oveqd 7368 . . . . . 6 (𝑀 = π‘Š β†’ (𝑝(distβ€˜π‘€)π‘₯) = (𝑝𝐷π‘₯))
1615eqeq1d 2739 . . . . 5 (𝑀 = π‘Š β†’ ((𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ ↔ (𝑝𝐷π‘₯) = π‘Ÿ))
178, 16rabeqbidv 3422 . . . 4 (𝑀 = π‘Š β†’ {𝑝 ∈ (Baseβ€˜π‘€) ∣ (𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ} = {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ})
188, 9, 17mpoeq123dv 7426 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ (Baseβ€˜π‘€), π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ (Baseβ€˜π‘€) ∣ (𝑝(distβ€˜π‘€)π‘₯) = π‘Ÿ}) = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
19 elex 3461 . . 3 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
20 fvex 6852 . . . . . 6 (Baseβ€˜π‘Š) ∈ V
215, 20eqeltri 2834 . . . . 5 𝐡 ∈ V
22 ovex 7384 . . . . 5 (0[,]+∞) ∈ V
2321, 22mpoex 8004 . . . 4 (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}) ∈ V
2423a1i 11 . . 3 (π‘Š ∈ 𝑉 β†’ (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}) ∈ V)
253, 18, 19, 24fvmptd3 6968 . 2 (π‘Š ∈ 𝑉 β†’ (Sphereβ€˜π‘Š) = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
262, 25eqtrd 2777 1 (π‘Š ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡, π‘Ÿ ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐡 ∣ (𝑝𝐷π‘₯) = π‘Ÿ}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {crab 3405  Vcvv 3443  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  0cc0 11009  +∞cpnf 11144  [,]cicc 13221  Basecbs 17043  distcds 17102  Spherecsph 46715
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-sph 46717
This theorem is referenced by:  sphere  46734  rrxsphere  46735
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