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Theorem blval 23892
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 ∈ (∞Metβ€˜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 blval
Dummy variables π‘Ÿ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 blfval 23890 . . 3 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (ballβ€˜π·) = (𝑦 ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {π‘₯ ∈ 𝑋 ∣ (𝑦𝐷π‘₯) < π‘Ÿ}))
213ad2ant1 1134 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ (ballβ€˜π·) = (𝑦 ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {π‘₯ ∈ 𝑋 ∣ (𝑦𝐷π‘₯) < π‘Ÿ}))
3 simprl 770 . . . . 5 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ 𝑦 = 𝑃)
43oveq1d 7424 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ (𝑦𝐷π‘₯) = (𝑃𝐷π‘₯))
5 simprr 772 . . . 4 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ π‘Ÿ = 𝑅)
64, 5breq12d 5162 . . 3 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ ((𝑦𝐷π‘₯) < π‘Ÿ ↔ (𝑃𝐷π‘₯) < 𝑅))
76rabbidv 3441 . 2 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑦 = 𝑃 ∧ π‘Ÿ = 𝑅)) β†’ {π‘₯ ∈ 𝑋 ∣ (𝑦𝐷π‘₯) < π‘Ÿ} = {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅})
8 simp2 1138 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ 𝑃 ∈ 𝑋)
9 simp3 1139 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ 𝑅 ∈ ℝ*)
10 elfvdm 6929 . . . 4 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ dom ∞Met)
11103ad2ant1 1134 . . 3 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ 𝑋 ∈ dom ∞Met)
12 rabexg 5332 . . 3 (𝑋 ∈ dom ∞Met β†’ {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅} ∈ V)
1311, 12syl 17 . 2 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) β†’ {π‘₯ ∈ 𝑋 ∣ (𝑃𝐷π‘₯) < 𝑅} ∈ V)
142, 7, 8, 9, 13ovmpod 7560 1 ((𝐷 ∈ (∞Metβ€˜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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  β„*cxr 11247   < clt 11248  βˆžMetcxmet 20929  ballcbl 20931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-xr 11252  df-psmet 20936  df-xmet 20937  df-bl 20939
This theorem is referenced by:  elbl  23894  metss2lem  24020  stdbdbl  24026  nmhmcn  24636  lgamucov  26542  isbnd3  36652
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