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Theorem blfvalps 23881
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 20932 . 2 ball = (𝑑 ∈ V ↦ (π‘₯ ∈ dom dom 𝑑, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ}))
2 dmeq 5902 . . . . 5 (𝑑 = 𝐷 β†’ dom 𝑑 = dom 𝐷)
32dmeqd 5904 . . . 4 (𝑑 = 𝐷 β†’ dom dom 𝑑 = dom dom 𝐷)
4 psmetdmdm 23803 . . . . 5 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 = dom dom 𝐷)
54eqcomd 2739 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ dom dom 𝐷 = 𝑋)
63, 5sylan9eqr 2795 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ dom dom 𝑑 = 𝑋)
7 eqidd 2734 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ℝ* = ℝ*)
8 simpr 486 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ 𝑑 = 𝐷)
98oveqd 7423 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ (π‘₯𝑑𝑦) = (π‘₯𝐷𝑦))
109breq1d 5158 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ ((π‘₯𝑑𝑦) < π‘Ÿ ↔ (π‘₯𝐷𝑦) < π‘Ÿ))
116, 10rabeqbidv 3450 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑑 = 𝐷) β†’ {𝑦 ∈ dom dom 𝑑 ∣ (π‘₯𝑑𝑦) < π‘Ÿ} = {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ})
126, 7, 11mpoeq123dv 7481 . 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 3493 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝐷 ∈ V)
14 ssrab2 4077 . . . . . 6 {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} βŠ† 𝑋
15 elfvdm 6926 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 ∈ dom PsMet)
1615adantr 482 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*)) β†’ 𝑋 ∈ dom PsMet)
17 elpw2g 5344 . . . . . . 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 3201 . . . 4 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘Ÿ ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ} ∈ 𝒫 𝑋)
21 eqid 2733 . . . . 5 (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) = (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ})
2221fmpo 8051 . . . 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 12968 . . . 4 ℝ* ∈ V
25 xpexg 7734 . . . 4 ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) β†’ (𝑋 Γ— ℝ*) ∈ V)
2615, 24, 25sylancl 587 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝑋 Γ— ℝ*) ∈ V)
2715pwexd 5377 . . 3 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝒫 𝑋 ∈ V)
28 fex2 7921 . . 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 1372 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (π‘₯ ∈ 𝑋, π‘Ÿ ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (π‘₯𝐷𝑦) < π‘Ÿ}) ∈ V)
301, 12, 13, 29fvmptd2 7004 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475   βŠ† wss 3948  π’« cpw 4602   class class class wbr 5148   Γ— cxp 5674  dom cdm 5676  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  β„*cxr 11244   < clt 11245  PsMetcpsmet 20921  ballcbl 20924
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-map 8819  df-xr 11249  df-psmet 20929  df-bl 20932
This theorem is referenced by:  blfval  23882  blvalps  23883  blfps  23904
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