Theorem sphere 49235

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 49234	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
5	4	3ad2ant1 1134	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
6		oveq2 7368	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑝𝐷𝑥) = (𝑝𝐷𝑋))
7		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
8	6, 7	eqeqan12d 2751	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑝𝐷𝑥) = 𝑟 ↔ (𝑝𝐷𝑋) = 𝑅))
9	8	rabbidv 3397	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
10	9	adantl 481	. 2 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) ∧ (𝑥 = 𝑋 ∧ 𝑟 = 𝑅)) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
11		simp2 1138	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑋 ∈ 𝐵)
12		simp3 1139	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑅 ∈ (0[,]+∞))
13	1	fvexi 6848	. . . 4 ⊢ 𝐵 ∈ V
14	13	rabex 5276	. . 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 7512	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  0cc0 11029  +∞cpnf 11167  [,]cicc 13292  Basecbs 17170  distcds 17220  Spherecsph 49216

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-sph 49218

This theorem is referenced by:  rrxsphere  49236