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Theorem blfps 22432
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 3828 . . . . . 6 {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
2 elfvdm 6382 . . . . . . 7 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
3 elpw2g 4976 . . . . . . 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 248 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
65a1d 25 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((𝑥𝑋𝑟 ∈ ℝ*) → {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋))
76ralrimivv 3108 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
8 eqid 2760 . . . 4 (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
98fmpt2 7406 . . 3 (∀𝑥𝑋𝑟 ∈ ℝ* {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
107, 9sylib 208 . 2 (𝐷 ∈ (PsMet‘𝑋) → (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
11 blfvalps 22409 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
1211feq1d 6191 . 2 (𝐷 ∈ (PsMet‘𝑋) → ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ↔ (𝑥𝑋, 𝑟 ∈ ℝ* ↦ {𝑦𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋))
1310, 12mpbird 247 1 (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wral 3050  {crab 3054  wss 3715  𝒫 cpw 4302   class class class wbr 4804   × cxp 5264  dom cdm 5266  wf 6045  cfv 6049  (class class class)co 6814  cmpt2 6816  *cxr 10285   < clt 10286  PsMetcpsmet 19952  ballcbl 19955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-xr 10290  df-psmet 19960  df-bl 19963
This theorem is referenced by:  blrnps  22434  blelrnps  22442  unirnblps  22445
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