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Theorem blfvalps 23524
Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
blfvalps (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Distinct variable groups:   𝑥,𝑟,𝑦,𝐷   𝑋,𝑟,𝑥,𝑦

Proof of Theorem blfvalps
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-bl 20580 . 2 ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
2 dmeq 5806 . . . . 5 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
32dmeqd 5808 . . . 4 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
4 psmetdmdm 23446 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
54eqcomd 2744 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
63, 5sylan9eqr 2800 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
7 eqidd 2739 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
8 simpr 485 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
98oveqd 7285 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
109breq1d 5084 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
116, 10rabeqbidv 3418 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
126, 7, 11mpoeq123dv 7341 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
13 elex 3448 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
14 ssrab2 4013 . . . . . 6 {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
15 elfvdm 6799 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
1615adantr 481 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
17 elpw2g 5267 . . . . . . 7 (𝑋 ∈ dom PsMet → ({𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
1816, 17syl 17 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → ({𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
1914, 18mpbiri 257 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
2019ralrimivva 3120 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
21 eqid 2738 . . . . 5 (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
2221fmpo 7898 . . . 4 (∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
2320, 22sylib 217 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
24 xrex 12715 . . . 4 * ∈ V
25 xpexg 7591 . . . 4 ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
2615, 24, 25sylancl 586 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
2715pwexd 5301 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
28 fex2 7771 . . 3 (((𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ (𝑋 × ℝ*) ∈ V ∧ 𝒫 𝑋 ∈ V) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
2923, 26, 27, 28syl3anc 1370 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
301, 12, 13, 29fvmptd2 6876 1 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3430  wss 3887  𝒫 cpw 4534   class class class wbr 5074   × cxp 5583  dom cdm 5585  wf 6423  cfv 6427  (class class class)co 7268  cmpo 7270  *cxr 10996   < clt 10997  PsMetcpsmet 20569  ballcbl 20572
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579  ax-cnex 10915  ax-resscn 10916
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-fv 6435  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7821  df-2nd 7822  df-map 8605  df-xr 11001  df-psmet 20577  df-bl 20580
This theorem is referenced by:  blfval  23525  blvalps  23526  blfps  23547
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