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Theorem dffn5im 5435
Description: Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 5131 and dmmptss 5005. (Contributed by Jim Kingdon, 31-Dec-2018.)
Assertion
Ref Expression
dffn5im  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem dffn5im
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrel 5191 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 4960 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 121 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5195 . . . . . . 7  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 114 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 391 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2119 . . . . . . 7  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
8 fnbrfvb 5430 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
97, 8syl5bb 191 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
109pm5.32da 447 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 190 . . . 4  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
1211opabbidv 3964 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
133, 12eqtrd 2150 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
14 df-mpt 3961 . 2  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
1513, 14syl6eqr 2168 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   class class class wbr 3899   {copab 3958    |-> cmpt 3959   Rel wrel 4514    Fn wfn 5088   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fnrnfv  5436  feqmptd  5442  dffn5imf  5444  eqfnfv  5486  fndmin  5495  fcompt  5558  resfunexg  5609  eufnfv  5616  fnovim  5847  offveqb  5969  caofinvl  5972  oprabco  6082  df1st2  6084  df2nd2  6085  xpen  6707  cnmpt1st  12384  cnmpt2nd  12385
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