Theorem sphere 47521

Description: A sphere with center 𝑋 and radius 𝑅 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
sphere	⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})

Distinct variable groups:   𝐵,𝑝   𝑊,𝑝   𝑅,𝑝   𝑋,𝑝

Allowed substitution hints:   𝐷(𝑝)   𝑆(𝑝)   𝑉(𝑝)

Proof of Theorem sphere

Dummy variables 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		spheres.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
2		spheres.l	. . . 4 ⊢ 𝑆 = (Sphere‘𝑊)
3		spheres.d	. . . 4 ⊢ 𝐷 = (dist‘𝑊)
4	1, 2, 3	spheres 47520	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
5	4	3ad2ant1 1133	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
6		oveq2 7419	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑝𝐷𝑥) = (𝑝𝐷𝑋))
7		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
8	6, 7	eqeqan12d 2746	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑝𝐷𝑥) = 𝑟 ↔ (𝑝𝐷𝑋) = 𝑅))
9	8	rabbidv 3440	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
10	9	adantl 482	. 2 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) ∧ (𝑥 = 𝑋 ∧ 𝑟 = 𝑅)) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
11		simp2 1137	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑋 ∈ 𝐵)
12		simp3 1138	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑅 ∈ (0[,]+∞))
13	1	fvexi 6905	. . . 4 ⊢ 𝐵 ∈ V
14	13	rabex 5332	. . 3 ⊢ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅} ∈ V
15	14	a1i 11	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅} ∈ V)
16	5, 10, 11, 12, 15	ovmpod 7562	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  ‘cfv 6543  (class class class)co 7411   ∈ cmpo 7413  0cc0 11112  +∞cpnf 11249  [,]cicc 13331  Basecbs 17148  distcds 17210  Spherecsph 47502

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-sph 47504

This theorem is referenced by:  rrxsphere  47522