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Theorem blfvalps 24210
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 21222 . 2 ball = (𝑑 ∈ V ↦ (π‘₯ ∈ dom dom 𝑑, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ}))
2 dmeq 5893 . . . . 5 (𝑑 = 𝐷 β†’ dom 𝑑 = dom 𝐷)
32dmeqd 5895 . . . 4 (𝑑 = 𝐷 β†’ dom dom 𝑑 = dom dom 𝐷)
4 psmetdmdm 24132 . . . . 5 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 = dom dom 𝐷)
54eqcomd 2730 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom dom 𝐷 = 𝑋)
63, 5sylan9eqr 2786 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom dom 𝑑 = 𝑋)
7 eqidd 2725 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ℝ* = ℝ*)
8 simpr 484 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ 𝑑 = 𝐷)
98oveqd 7418 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (π‘₯𝑑𝑦) = (π‘₯𝐷𝑦))
109breq1d 5148 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((π‘₯𝑑𝑦) < π‘Ÿ ↔ (π‘₯𝐷𝑦) < π‘Ÿ))
116, 10rabeqbidv 3441 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ} = {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ})
126, 7, 11mpoeq123dv 7476 . 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 3485 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷 ∈ V)
14 ssrab2 4069 . . . . . 6 {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} βŠ† 𝑋
15 elfvdm 6918 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 ∈ dom PsMet)
1615adantr 480 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*)) β†’ 𝑋 ∈ dom PsMet)
17 elpw2g 5334 . . . . . . 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 258 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*)) β†’ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋)
2019ralrimivva 3192 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘Ÿ ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋)
21 eqid 2724 . . . . 5 (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) = (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ})
2221fmpo 8047 . . . 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 12967 . . . 4 ℝ* ∈ V
25 xpexg 7730 . . . 4 ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) β†’ (𝑋 Γ— ℝ*) ∈ V)
2615, 24, 25sylancl 585 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝑋 Γ— ℝ*) ∈ V)
2715pwexd 5367 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝒫 𝑋 ∈ V)
28 fex2 7917 . . 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 1368 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) ∈ V)
301, 12, 13, 29fvmptd2 6996 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  {crab 3424  Vcvv 3466   βŠ† wss 3940  π’« cpw 4594   class class class wbr 5138   Γ— cxp 5664  dom cdm 5666  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403  β„*cxr 11243   < clt 11244  PsMetcpsmet 21211  ballcbl 21214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8817  df-xr 11248  df-psmet 21219  df-bl 21222
This theorem is referenced by:  blfval  24211  blvalps  24212  blfps  24233
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