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Theorem blelrn 20683
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 20673 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D ) : ( X  X.  RR* )
--> ~P X )
2 ffn 5731 . . 3  |-  ( (
ball `  D ) : ( X  X.  RR* ) --> ~P X  -> 
( ball `  D )  Fn  ( X  X.  RR* ) )
31, 2syl 16 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( ball `  D )  Fn  ( X  X.  RR* ) )
4 fnovrn 6434 . 2  |-  ( ( ( ball `  D
)  Fn  ( X  X.  RR* )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  e. 
ran  ( ball `  D
) )
53, 4syl3an1 1261 1  |-  ( ( D  e.  ( *Met `  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 973    e. wcel 1767   ~Pcpw 4010    X. cxp 4997   ran crn 5000    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   RR*cxr 9627   *Metcxmt 18202   ballcbl 18204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-xr 9632  df-psmet 18210  df-xmet 18211  df-bl 18213
This theorem is referenced by:  unirnbl  20686  blssex  20693  blopn  20766  blcld  20771  metss  20774  metcnp3  20806  dscopn  20857  ioo2blex  21062
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