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Theorem fnrnfv 5585
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 5584 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 rneq 4920 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  ran  F  =  ran  (
x  e.  A  |->  ( F `  x ) ) )
31, 2sylbi 187 . 2  |-  ( F  Fn  A  ->  ran  F  =  ran  ( x  e.  A  |->  ( F `
 x ) ) )
4 eqid 2296 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
54rnmpt 4941 . 2  |-  ran  (
x  e.  A  |->  ( F `  x ) )  =  { y  |  E. x  e.  A  y  =  ( F `  x ) }
63, 5syl6eq 2344 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 1632   {cab 2282   E.wrex 2557    e. cmpt 4093   ran crn 4706    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  fvelrnb  5586  fniinfv  5597  dffo3  5691  fniunfv  5789  fnrnov  6009  pwcfsdom  8221  hauscmplem  17149  grpoinvf  20923  meascnbl  23561  fargshiftfo  28383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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