ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffn5im Unicode version

Theorem dffn5im 5467
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 5161 and dmmptss 5035. (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 5221 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 4990 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 121 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5225 . . . . . . 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 2141 . . . . . . 7  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
8 fnbrfvb 5462 . . . . . . 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 3994 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
133, 12eqtrd 2172 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
14 df-mpt 3991 . 2  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
1513, 14syl6eqr 2190 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   class class class wbr 3929   {copab 3988    |-> cmpt 3989   Rel wrel 4544    Fn wfn 5118   ` cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  fnrnfv  5468  feqmptd  5474  dffn5imf  5476  eqfnfv  5518  fndmin  5527  fcompt  5590  resfunexg  5641  eufnfv  5648  fnovim  5879  offveqb  6001  caofinvl  6004  oprabco  6114  df1st2  6116  df2nd2  6117  xpen  6739  cnmpt1st  12466  cnmpt2nd  12467
  Copyright terms: Public domain W3C validator