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Theorem blelrnps 13134
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 13124 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D ) : ( X  X.  RR* ) --> ~P X )
21ffnd 5346 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( ball `  D )  Fn  ( X  X.  RR* ) )
3 fnovrn 5997 . 2  |-  ( ( ( ball `  D
)  Fn  ( X  X.  RR* )  /\  P  e.  X  /\  R  e. 
RR* )  ->  ( P ( ball `  D
) R )  e. 
ran  ( ball `  D
) )
42, 3syl3an1 1266 1  |-  ( ( D  e.  (PsMet `  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 973    e. wcel 2141   ~Pcpw 3564    X. cxp 4607   ran crn 4610    Fn wfn 5191   ` cfv 5196  (class class class)co 5850   RR*cxr 7940  PsMetcpsmet 12694   ballcbl 12697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-map 6624  df-pnf 7943  df-mnf 7944  df-xr 7945  df-psmet 12702  df-bl 12705
This theorem is referenced by:  unirnblps  13137  blssexps  13144
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