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Theorem blfvalps 23688
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 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·) = (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}))
Distinct variable groups:   π‘₯,π‘Ÿ,𝑦,𝐷   𝑋,π‘Ÿ,π‘₯,𝑦

Proof of Theorem blfvalps
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 df-bl 20744 . 2 ball = (𝑑 ∈ V ↦ (π‘₯ ∈ dom dom 𝑑, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ}))
2 dmeq 5857 . . . . 5 (𝑑 = 𝐷 β†’ dom 𝑑 = dom 𝐷)
32dmeqd 5859 . . . 4 (𝑑 = 𝐷 β†’ dom dom 𝑑 = dom dom 𝐷)
4 psmetdmdm 23610 . . . . 5 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 = dom dom 𝐷)
54eqcomd 2743 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom dom 𝐷 = 𝑋)
63, 5sylan9eqr 2799 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom dom 𝑑 = 𝑋)
7 eqidd 2738 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ℝ* = ℝ*)
8 simpr 485 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ 𝑑 = 𝐷)
98oveqd 7368 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (π‘₯𝑑𝑦) = (π‘₯𝐷𝑦))
109breq1d 5113 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((π‘₯𝑑𝑦) < π‘Ÿ ↔ (π‘₯𝐷𝑦) < π‘Ÿ))
116, 10rabeqbidv 3422 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ} = {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ})
126, 7, 11mpoeq123dv 7426 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (π‘₯ ∈ dom dom 𝑑, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ}) = (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}))
13 elex 3461 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷 ∈ V)
14 ssrab2 4035 . . . . . 6 {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} βŠ† 𝑋
15 elfvdm 6876 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 ∈ dom PsMet)
1615adantr 481 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*)) β†’ 𝑋 ∈ dom PsMet)
17 elpw2g 5299 . . . . . . 7 (𝑋 ∈ dom PsMet β†’ ({𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} βŠ† 𝑋))
1816, 17syl 17 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*)) β†’ ({𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} βŠ† 𝑋))
1914, 18mpbiri 257 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*)) β†’ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋)
2019ralrimivva 3195 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘Ÿ ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋)
21 eqid 2737 . . . . 5 (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) = (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ})
2221fmpo 7992 . . . 4 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘Ÿ ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋 ↔ (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
2320, 22sylib 217 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋)
24 xrex 12866 . . . 4 ℝ* ∈ V
25 xpexg 7676 . . . 4 ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) β†’ (𝑋 Γ— ℝ*) ∈ V)
2615, 24, 25sylancl 586 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝑋 Γ— ℝ*) ∈ V)
2715pwexd 5332 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝒫 𝑋 ∈ V)
28 fex2 7862 . . 3 (((π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}):(𝑋 Γ— ℝ*)βŸΆπ’« 𝑋 ∧ (𝑋 Γ— ℝ*) ∈ V ∧ 𝒫 𝑋 ∈ V) β†’ (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) ∈ V)
2923, 26, 27, 28syl3anc 1371 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) ∈ V)
301, 12, 13, 29fvmptd2 6953 1 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (ballβ€˜π·) = (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3062  {crab 3405  Vcvv 3443   βŠ† wss 3908  π’« cpw 4558   class class class wbr 5103   Γ— cxp 5629  dom cdm 5631  βŸΆwf 6489  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  β„*cxr 11146   < clt 11147  PsMetcpsmet 20733  ballcbl 20736
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-map 8725  df-xr 11151  df-psmet 20741  df-bl 20744
This theorem is referenced by:  blfval  23689  blvalps  23690  blfps  23711
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