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Theorem dffn5 5584
Description: Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dffn5  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem dffn5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrel 5358 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5141 . . . . 5  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 188 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5362 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 423 . . . . . . 7  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 616 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2298 . . . . . . . 8  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
8 fnbrfvb 5579 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
97, 8syl5bb 248 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
109pm5.32da 622 . . . . . 6  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 247 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
1211opabbidv 4098 . . . 4  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
133, 12eqtrd 2328 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
14 df-mpt 4095 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
1513, 14syl6eqr 2346 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
16 fvex 5555 . . . 4  |-  ( F `
 x )  e. 
_V
17 eqid 2296 . . . 4  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
1816, 17fnmpti 5388 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  Fn  A
19 fneq1 5349 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  -> 
( F  Fn  A  <->  ( x  e.  A  |->  ( F `  x ) )  Fn  A ) )
2018, 19mpbiri 224 . 2  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  F  Fn  A )
2115, 20impbii 180 1  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   {copab 4092    e. cmpt 4093   Rel wrel 4710    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  fnrnfv  5585  feqmptd  5591  dffn5f  5593  eqfnfv  5638  fndmin  5648  fcompt  5710  resfunexg  5753  eufnfv  5768  fnov  5968  offveqb  6115  caofinvl  6120  oprabco  6219  df1st2  6221  df2nd2  6222  curry1  6226  curry2  6229  resixpfo  6870  pw2f1olem  6982  marypha2lem3  7206  seqof  11119  prmrec  12985  prdsbascl  13398  xpsaddlem  13493  xpsvsca  13497  oppccatid  13638  fuclid  13856  fucrid  13857  curfuncf  14028  yonedainv  14071  yonffthlem  14072  prdsidlem  14420  pws0g  14424  prdsinvlem  14619  staffn  15630  prdslmodd  15742  cnmpt1st  17378  cnmpt2nd  17379  ptunhmeo  17515  xpsxmetlem  17959  xpsmet  17962  itg2split  19120  pserulm  19814  pserdvlem2  19820  logcn  20010  emcllem5  20309  fcomptf  23245  esumpcvgval  23461  dstfrvclim1  23693  ptpcon  23779  ovoliunnfl  25001  fnovpop  25141  fnopabco2b  25474  fnopabco  26491  upixp  26506  prdsbnd  26620  prdstotbnd  26621  prdsbnd2  26622  fgraphopab  27632  expgrowthi  27653  expgrowth  27655  dvcosre  27844  stoweidlem20  27872
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-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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