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Theorem fnrnfv 5364
Description: The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fnrnfv  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
Distinct variable groups:    x, y, A   
x, F, y

Proof of Theorem fnrnfv
StepHypRef Expression
1 dffn5im 5363 . . 3  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
21rneqd 4677 . 2  |-  ( F  Fn  A  ->  ran  F  =  ran  ( x  e.  A  |->  ( F `
 x ) ) )
3 eqid 2089 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
43rnmpt 4696 . 2  |-  ran  (
x  e.  A  |->  ( F `  x ) )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6eq 2137 1  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   {cab 2075   E.wrex 2361    |-> cmpt 3905   ran crn 4452    Fn wfn 5023   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036
This theorem is referenced by:  fvelrnb  5365  fniinfv  5375  dffo3  5460  fniunfv  5555  fnrnov  5804
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