Theorem sphere 49803

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 49802	. . 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 2775	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑝𝐷𝑥) = 𝑟 ↔ (𝑝𝐷𝑋) = 𝑅))
9	8	rabbidv 3420	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
10	9	adantl 487	. 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 6891	. . . 4 ⊢ 𝐵 ∈ V
14	13	rabex 5300	. . 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})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  0cc0 11181  +∞cpnf 11321  [,]cicc 13460  Basecbs 17367  distcds 17417  Spherecsph 49784

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-sph 49786

This theorem is used by:  rrxsphere  49804