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Theorem fnrnfv 5918
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 dffn5 5917 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 rneq 5071 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  ran  F  =  ran  (
x  e.  A  |->  ( F `  x ) ) )
31, 2sylbi 198 . 2  |-  ( F  Fn  A  ->  ran  F  =  ran  ( x  e.  A  |->  ( F `
 x ) ) )
4 eqid 2420 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
54rnmpt 5091 . 2  |-  ran  (
x  e.  A  |->  ( F `  x ) )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
63, 5syl6eq 2477 1  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {cab 2405   E.wrex 2774    |-> cmpt 4475   ran crn 4846    Fn wfn 5587   ` cfv 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600
This theorem is referenced by:  fvelrnb  5919  fniinfv  5931  dffo3  6043  fniunfv  6158  fnrnov  6447  pwcfsdom  8997  hauscmplem  20345  fargshiftfo  25237  grpoinvf  25839  fpwrelmapffslem  28186  meascnbl  28906  omssubadd  28987  dffo3f  37106  rnfdmpr  38427
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