Theorem blval 24330

Description: The ball around a point 𝑃 is the set of all points whose distance from 𝑃 is less than the ball's radius 𝑅. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Assertion

Ref	Expression
blval	⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})

Distinct variable groups:   𝑥,𝑃   𝑥,𝐷   𝑥,𝑅   𝑥,𝑋

Proof of Theorem blval

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

Step	Hyp	Ref	Expression
1		blfval 24328	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑦 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (ball‘𝐷) = (𝑦 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟}))
3		simprl 770	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → 𝑦 = 𝑃)
4	3	oveq1d 7373	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → (𝑦𝐷𝑥) = (𝑃𝐷𝑥))
5		simprr 772	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → 𝑟 = 𝑅)
6	4, 5	breq12d 5111	. . 3 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → ((𝑦𝐷𝑥) < 𝑟 ↔ (𝑃𝐷𝑥) < 𝑅))
7	6	rabbidv 3406	. 2 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ 𝑟 = 𝑅)) → {𝑥 ∈ 𝑋 ∣ (𝑦𝐷𝑥) < 𝑟} = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})
8		simp2 1137	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑃 ∈ 𝑋)
9		simp3 1138	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑅 ∈ ℝ*)
10		elfvdm 6868	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
11	10	3ad2ant1 1133	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝑋 ∈ dom ∞Met)
12		rabexg 5282	. . 3 ⊢ (𝑋 ∈ dom ∞Met → {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
13	11, 12	syl 17	. 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ∈ V)
14	2, 7, 8, 9, 13	ovmpod 7510	1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ℝ^*cxr 11165   < clt 11166  ∞Metcxmet 21294  ballcbl 21296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  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-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-xr 11170  df-psmet 21301  df-xmet 21302  df-bl 21304

This theorem is referenced by:  elbl  24332  metss2lem  24455  stdbdbl  24461  nmhmcn  25076  lgamucov  27004  isbnd3  37981