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Theorem blvalps 24111
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blvalps ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)𝑅) = {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅})
Distinct variable groups:   π‘₯,𝑃   π‘₯,𝐷   π‘₯,𝑅   π‘₯,𝑋
Proof of Theorem blvalps
Dummy variables π‘Ÿ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfvalps 24109 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·) = (𝑦 ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {π‘₯ ∈ 𝑋 ∣ (𝑦𝐷π‘₯) < π‘Ÿ}))
213ad2ant1 1133 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ (ballβ€˜π·) = (𝑦 ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {π‘₯ ∈ 𝑋 ∣ (𝑦𝐷π‘₯) < π‘Ÿ}))
3 simprl 769 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ 𝑦 = 𝑃)
43oveq1d 7426 . . . 4 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ (𝑦𝐷π‘₯) = (𝑃𝐷π‘₯))
5 simprr 771 . . . 4 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ π‘Ÿ = 𝑅)
64, 5breq12d 5161 . . 3 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ ((𝑦𝐷π‘₯) < π‘Ÿ ↔ (𝑃𝐷π‘₯) < 𝑅))
76rabbidv 3440 . 2 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ {π‘₯ ∈ 𝑋 ∣ (𝑦𝐷π‘₯) < π‘Ÿ} = {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅})
8 simp2 1137 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ 𝑃 ∈ 𝑋)
9 simp3 1138 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ 𝑅 ∈ ℝ*)
10 elfvdm 6928 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 ∈ dom PsMet)
11103ad2ant1 1133 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ 𝑋 ∈ dom PsMet)
12 rabexg 5331 . . 3 (𝑋 ∈ dom PsMet β†’ {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅} ∈ V)
1311, 12syl 17 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅} ∈ V)
142, 7, 8, 9, 13ovmpod 7562 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)𝑅) = {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   class class class wbr 5148  dom cdm 5676  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  β„*cxr 11251   < clt 11252  PsMetcpsmet 21128  ballcbl 21131
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-xr 11256  df-psmet 21136  df-bl 21139
This theorem is referenced by:  elblps  24113  blval2  24291
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