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Theorem fnrnfv 5774
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 5773 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 rneq 5096 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  ran  F  =  ran  (
x  e.  A  |->  ( F `  x ) ) )
31, 2sylbi 189 . 2  |-  ( F  Fn  A  ->  ran  F  =  ran  ( x  e.  A  |->  ( F `
 x ) ) )
4 eqid 2437 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
54rnmpt 5117 . 2  |-  ran  (
x  e.  A  |->  ( F `  x ) )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
63, 5syl6eq 2485 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 1653   {cab 2423   E.wrex 2707    e. cmpt 4267   ran crn 4880    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  fvelrnb  5775  fniinfv  5786  dffo3  5885  fniunfv  5995  fnrnov  6220  pwcfsdom  8459  hauscmplem  17470  fargshiftfo  21626  grpoinvf  21829  meascnbl  24574  rnfdmpr  28084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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