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Theorem blfps 23016
Description: Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blfps (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)

Proof of Theorem blfps
Dummy variables 𝑥 𝑟 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 4010 . . . . . 6 {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
2 elfvdm 6681 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
3 elpw2g 5214 . . . . . . 7 (𝑋 ∈ dom PsMet → ({𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
42, 3syl 17 . . . . . 6 (𝐷 ∈ (PsMet‘𝑋) → ({𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
51, 4mpbiri 261 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
65a1d 25 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((𝑥𝑋𝑟 ∈ ℝ*) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋))
76ralrimivv 3158 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
8 eqid 2801 . . . 4 (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
98fmpo 7752 . . 3 (∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
107, 9sylib 221 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
11 blfvalps 22993 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
1211feq1d 6476 . 2 (𝐷 ∈ (PsMet‘𝑋) → ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋))
1310, 12mpbird 260 1 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2112  wral 3109  {crab 3113  wss 3884  𝒫 cpw 4500   class class class wbr 5033   × cxp 5521  dom cdm 5523  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  *cxr 10667   < clt 10668  PsMetcpsmet 20078  ballcbl 20081
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-xr 10672  df-psmet 20086  df-bl 20089
This theorem is referenced by:  blrnps  23018  blelrnps  23026  unirnblps  23029
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