Theorem sphere 49504

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 49503	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
5	4	3ad2ant1 1151	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
6		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑝𝐷𝑥) = (𝑝𝐷𝑋))
7		id 23	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
8	6, 7	eqeqan12d 2777	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑝𝐷𝑥) = 𝑟 ↔ (𝑝𝐷𝑋) = 𝑅))
9	8	rabbidv 3423	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
10	9	adantl 486	. 2 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) ∧ (𝑥 = 𝑋 ∧ 𝑟 = 𝑅)) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
11		simp2 1155	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑋 ∈ 𝐵)
12		simp3 1156	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑅 ∈ (0[,]+∞))
13	1	fvexi 6897	. . . 4 ⊢ 𝐵 ∈ V
14	13	rabex 5311	. . 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 7564	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  0cc0 11101  +∞cpnf 11241  [,]cicc 13376  Basecbs 17270  distcds 17320  Spherecsph 49485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-sph 49487

This theorem is referenced by:  rrxsphere  49505