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Theorem blfval 22986
 Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Proof shortened by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
blfval (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Distinct variable groups:   𝑥,𝑟,𝑦,𝐷   𝑋,𝑟,𝑥,𝑦

Proof of Theorem blfval
StepHypRef Expression
1 xmetpsmet 22950 . 2 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
2 blfvalps 22985 . 2 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
31, 2syl 17 1 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108  {crab 3140   class class class wbr 5057  ‘cfv 6348  (class class class)co 7148   ∈ cmpo 7150  ℝ*cxr 10666   < clt 10667  PsMetcpsmet 20521  ∞Metcxmet 20522  ballcbl 20524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-xr 10671  df-psmet 20529  df-xmet 20530  df-bl 20532 This theorem is referenced by:  blval  22988  blf  23009
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