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Theorem blelrn 22591
Description: A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Assertion
Ref Expression
blelrn ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))

Proof of Theorem blelrn
StepHypRef Expression
1 blf 22581 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
21ffnd 6278 . 2 (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷) Fn (𝑋 × ℝ*))
3 fnovrn 7068 . 2 (((ball‘𝐷) Fn (𝑋 × ℝ*) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
42, 3syl3an1 1208 1 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1113  wcel 2166  𝒫 cpw 4377   × cxp 5339  ran crn 5342   Fn wfn 6117  cfv 6122  (class class class)co 6904  *cxr 10389  ∞Metcxmet 20090  ballcbl 20092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-cnex 10307  ax-resscn 10308
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-1st 7427  df-2nd 7428  df-map 8123  df-xr 10394  df-psmet 20097  df-xmet 20098  df-bl 20100
This theorem is referenced by:  unirnbl  22594  blssex  22601  blopn  22674  blcld  22679  metss  22682  metcnp3  22714  dscopn  22747  ioo2blex  22966
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