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Theorem blelrnps 21211
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.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blelrnps  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  e. 
ran  ( ball `  D
) )

Proof of Theorem blelrnps
StepHypRef Expression
1 blfps 21201 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
2 ffn 5714 . . 3  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  -> 
( ball `  D )  Fn  ( X  X.  RR* ) )
31, 2syl 17 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  Fn  ( X  X.  RR* ) )
4 fnovrn 6431 . 2  |-  ( ( ( ball `  D
)  Fn  ( X  X.  RR* )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  e. 
ran  ( ball `  D
) )
53, 4syl3an1 1263 1  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  e. 
ran  ( ball `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    e. wcel 1842   ~Pcpw 3955    X. cxp 4821   ran crn 4824    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   RR*cxr 9657  PsMetcpsmet 18722   ballcbl 18725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-map 7459  df-xr 9662  df-psmet 18731  df-bl 18734
This theorem is referenced by:  unirnblps  21214  blssexps  21221
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