Theorem sphere 49333

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 49332	. . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
5	4	3ad2ant1 1145	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑆 = (𝑥 ∈ 𝐵, 𝑟 ∈ (0[,]+∞) ↦ {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟}))
6		oveq2 7400	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑝𝐷𝑥) = (𝑝𝐷𝑋))
7		id 22	. . . . 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 485	. 2 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) ∧ (𝑥 = 𝑋 ∧ 𝑟 = 𝑅)) → {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑥) = 𝑟} = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})
11		simp2 1149	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑋 ∈ 𝐵)
12		simp3 1150	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → 𝑅 ∈ (0[,]+∞))
13	1	fvexi 6877	. . . 4 ⊢ 𝐵 ∈ V
14	13	rabex 5294	. . 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 7544	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑅 ∈ (0[,]+∞)) → (𝑋𝑆𝑅) = {𝑝 ∈ 𝐵 ∣ (𝑝𝐷𝑋) = 𝑅})

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  0cc0 11070  +∞cpnf 11210  [,]cicc 13349  Basecbs 17228  distcds 17278  Spherecsph 49314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-sph 49316

This theorem is referenced by:  rrxsphere  49334