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Theorem dffn5im 5563
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 5256 and dmmptss 5127. (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 5316 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5082 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 122 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5320 . . . . . . 7  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 115 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 394 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2179 . . . . . . 7  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
8 fnbrfvb 5558 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
97, 8bitrid 192 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
109pm5.32da 452 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 191 . . . 4  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
1211opabbidv 4071 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
133, 12eqtrd 2210 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
14 df-mpt 4068 . 2  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
1513, 14eqtr4di 2228 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   class class class wbr 4005   {copab 4065    |-> cmpt 4066   Rel wrel 4633    Fn wfn 5213   ` cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226
This theorem is referenced by:  fnrnfv  5564  feqmptd  5571  dffn5imf  5573  eqfnfv  5615  fndmin  5625  fcompt  5688  resfunexg  5739  eufnfv  5749  fnovim  5985  offveqb  6104  caofinvl  6107  oprabco  6220  df1st2  6222  df2nd2  6223  xpen  6847  cnmpt1st  13827  cnmpt2nd  13828
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