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Theorem dffn5im 5557
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 5250 and dmmptss 5121. (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 5310 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5076 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 122 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5314 . . . . . . 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 5552 . . . . . . 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 4066 . . 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 4063 . 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 4000   {copab 4060    |-> cmpt 4061   Rel wrel 4628    Fn wfn 5207   ` cfv 5212
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 4118  ax-pow 4171  ax-pr 4206
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 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220
This theorem is referenced by:  fnrnfv  5558  feqmptd  5565  dffn5imf  5567  eqfnfv  5609  fndmin  5619  fcompt  5682  resfunexg  5733  eufnfv  5742  fnovim  5977  offveqb  6096  caofinvl  6099  oprabco  6212  df1st2  6214  df2nd2  6215  xpen  6839  cnmpt1st  13448  cnmpt2nd  13449
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