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Theorem blfvalps 22986
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 20533 . 2 ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
2 dmeq 5759 . . . . 5 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
32dmeqd 5761 . . . 4 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
4 psmetdmdm 22908 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
54eqcomd 2830 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
63, 5sylan9eqr 2881 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
7 eqidd 2825 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
8 simpr 488 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
98oveqd 7162 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
109breq1d 5062 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
116, 10rabeqbidv 3471 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
126, 7, 11mpoeq123dv 7218 . 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 3498 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
14 ssrab2 4041 . . . . . 6 {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
15 elfvdm 6690 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
1615adantr 484 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
17 elpw2g 5233 . . . . . . 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 261 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
2019ralrimivva 3186 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
21 eqid 2824 . . . . 5 (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
2221fmpo 7756 . . . 4 (∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
2320, 22sylib 221 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
24 xrex 12379 . . . 4 * ∈ V
25 xpexg 7463 . . . 4 ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
2615, 24, 25sylancl 589 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
2715pwexd 5267 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
28 fex2 7628 . . 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 1368 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
301, 12, 13, 29fvmptd2 6764 1 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  wss 3919  𝒫 cpw 4521   class class class wbr 5052   × cxp 5540  dom cdm 5542  wf 6339  cfv 6343  (class class class)co 7145  cmpo 7147  *cxr 10666   < clt 10667  PsMetcpsmet 20522  ballcbl 20525
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7679  df-2nd 7680  df-map 8398  df-xr 10671  df-psmet 20530  df-bl 20533
This theorem is referenced by:  blfval  22987  blvalps  22988  blfps  23009
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