MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blrn Structured version   Visualization version   GIF version

Theorem blrn 24434
Description: Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blrn (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
Distinct variable groups:   𝑥,𝑟,𝐴   𝐷,𝑟,𝑥   𝑋,𝑟,𝑥

Proof of Theorem blrn
StepHypRef Expression
1 blf 24432 . 2 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
2 ffn 6736 . 2 ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 → (ball‘𝐷) Fn (𝑋 × ℝ*))
3 ovelrn 7608 . 2 ((ball‘𝐷) Fn (𝑋 × ℝ*) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
41, 2, 33syl 18 1 (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥𝑋𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  wrex 3067  𝒫 cpw 4604   × cxp 5686  ran crn 5689   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  *cxr 11291  ∞Metcxmet 21366  ballcbl 21368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-xr 11296  df-psmet 21373  df-xmet 21374  df-bl 21376
This theorem is referenced by:  blss  24450  imasf1oxms  24517  prdsxmslem2  24557  blssioo  24830
  Copyright terms: Public domain W3C validator