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Theorem dffn5 5772
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 5543 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5322 . . . . 5  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 189 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5547 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 424 . . . . . . 7  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 617 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2438 . . . . . . . 8  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
8 fnbrfvb 5767 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
97, 8syl5bb 249 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
109pm5.32da 623 . . . . . 6  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 248 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
1211opabbidv 4271 . . . 4  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
133, 12eqtrd 2468 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
14 df-mpt 4268 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
1513, 14syl6eqr 2486 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
16 fvex 5742 . . . 4  |-  ( F `
 x )  e. 
_V
17 eqid 2436 . . . 4  |-  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( F `  x ) )
1816, 17fnmpti 5573 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  Fn  A
19 fneq1 5534 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  -> 
( F  Fn  A  <->  ( x  e.  A  |->  ( F `  x ) )  Fn  A ) )
2018, 19mpbiri 225 . 2  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  F  Fn  A )
2115, 20impbii 181 1  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   {copab 4265    e. cmpt 4266   Rel wrel 4883    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  fnrnfv  5773  feqmptd  5779  dffn5f  5781  eqfnfv  5827  fndmin  5837  fcompt  5904  resfunexg  5957  eufnfv  5972  fnov  6178  offveqb  6326  caofinvl  6331  oprabco  6431  df1st2  6433  df2nd2  6434  curry1  6438  curry2  6441  resixpfo  7100  pw2f1olem  7212  marypha2lem3  7442  seqof  11380  prmrec  13290  prdsbascl  13705  xpsaddlem  13800  xpsvsca  13804  oppccatid  13945  fuclid  14163  fucrid  14164  curfuncf  14335  yonedainv  14378  yonffthlem  14379  prdsidlem  14727  pws0g  14731  prdsinvlem  14926  staffn  15937  prdslmodd  16045  cnmpt1st  17700  cnmpt2nd  17701  ptunhmeo  17840  xpsxmetlem  18409  xpsmet  18412  itg2split  19641  pserulm  20338  pserdvlem2  20344  logcn  20538  emcllem5  20838  fcomptf  24077  esumpcvgval  24468  dstfrvclim1  24735  gamcvg2lem  24843  ptpcon  24920  ovoliunnfl  26248  voliunnfl  26250  fnopabco  26424  upixp  26431  prdsbnd  26502  prdstotbnd  26503  prdsbnd2  26504  fgraphopab  27506  expgrowthi  27527  expgrowth  27529  dvcosre  27717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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