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Theorem fnrnfv 5750
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 5749 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 rneq 5077 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  ran  F  =  ran  (
x  e.  A  |->  ( F `  x ) ) )
31, 2sylbi 195 . 2  |-  ( F  Fn  A  ->  ran  F  =  ran  ( x  e.  A  |->  ( F `
 x ) ) )
4 eqid 2443 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
54rnmpt 5097 . 2  |-  ran  (
x  e.  A  |->  ( F `  x ) )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
63, 5syl6eq 2491 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 1369   {cab 2429   E.wrex 2728    e. cmpt 4362   ran crn 4853    Fn wfn 5425   ` cfv 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5393  df-fun 5432  df-fn 5433  df-fv 5438
This theorem is referenced by:  fvelrnb  5751  fniinfv  5762  dffo3  5870  fniunfv  5976  fnrnov  6248  pwcfsdom  8759  hauscmplem  19021  fargshiftfo  23536  grpoinvf  23739  fpwrelmapffslem  26044  meascnbl  26645  rnfdmpr  30161
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