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Theorem blelrn 17967
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  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  e.  ran  ( ball `  D ) )

Proof of Theorem blelrn
StepHypRef Expression
1 blf 17961 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 ffn 5389 . . 3  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  -> 
( ball `  D )  Fn  ( X  X.  RR* ) )
31, 2syl 15 . 2  |-  ( D  e.  ( * Met `  X )  ->  ( ball `  D )  Fn  ( X  X.  RR* ) )
4 fnovrn 5995 . 2  |-  ( ( ( ball `  D
)  Fn  ( X  X.  RR* )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  e. 
ran  ( ball `  D
) )
53, 4syl3an1 1215 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  R  e.  RR* )  ->  ( P ( ball `  D ) R )  e.  ran  ( ball `  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684   ~Pcpw 3625    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RR*cxr 8866   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  unirnbl  17969  blssex  17973  blopn  18046  blcld  18051  metss  18054  metcnp3  18086  dscopn  18096  ioo2blex  18300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-xr 8871  df-xmet 16373  df-bl 16375
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