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Theorem blfvalps 22408
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 19956 . . 3 ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
21a1i 11 . 2 (𝐷 ∈ (PsMet‘𝑋) → ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟})))
3 dmeq 5461 . . . . 5 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
43dmeqd 5463 . . . 4 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5 psmetdmdm 22330 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
65eqcomd 2777 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
74, 6sylan9eqr 2827 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
8 eqidd 2772 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
9 simpr 471 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
109oveqd 6813 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
1110breq1d 4797 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
127, 11rabeqbidv 3345 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
137, 8, 12mpt2eq123dv 6868 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
14 elex 3364 . 2 (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
15 ssrab2 3836 . . . . . 6 {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
16 elfvdm 6363 . . . . . . . 8 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
1716adantr 466 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
18 elpw2g 4959 . . . . . . 7 (𝑋 ∈ dom PsMet → ({𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
1917, 18syl 17 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → ({𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
2015, 19mpbiri 248 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥𝑋𝑟 ∈ ℝ*)) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
2120ralrimivva 3120 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
22 eqid 2771 . . . . 5 (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
2322fmpt2 7391 . . . 4 (∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
2421, 23sylib 208 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
25 xrex 12032 . . . 4 * ∈ V
26 xpexg 7111 . . . 4 ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
2716, 25, 26sylancl 574 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
28 pwexg 4981 . . . 4 (𝑋 ∈ dom PsMet → 𝒫 𝑋 ∈ V)
2916, 28syl 17 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
30 fex2 7272 . . 3 (((𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ (𝑋 × ℝ*) ∈ V ∧ 𝒫 𝑋 ∈ V) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
3124, 27, 29, 30syl3anc 1476 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
322, 13, 14, 31fvmptd 6432 1 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4298   class class class wbr 4787  cmpt 4864   × cxp 5248  dom cdm 5250  wf 6026  cfv 6030  (class class class)co 6796  cmpt2 6798  *cxr 10279   < clt 10280  PsMetcpsmet 19945  ballcbl 19948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-map 8015  df-xr 10284  df-psmet 19953  df-bl 19956
This theorem is referenced by:  blfval  22409  blvalps  22410  blfps  22431
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