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Theorem dfafn5a 28022
Description: Representation of a function in terms of its values, analogous to dffn5 5568 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfafn5a  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem dfafn5a
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fnrel 5342 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5125 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 188 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5346 . . . . . . 7  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 423 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 616 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2285 . . . . . . 7  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
8 fnbrafvb 28016 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F''' x )  =  y  <->  x F
y ) )
97, 8syl5bb 248 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F''' x )  <->  x F
y ) )
109pm5.32da 622 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F''' x ) )  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 247 . . . 4  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F''' x ) ) ) )
1211opabbidv 4082 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
133, 12eqtrd 2315 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
14 df-mpt 4079 . 2  |-  ( x  e.  A  |->  ( F''' x ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) }
1513, 14syl6eqr 2333 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   {copab 4076    e. cmpt 4077   Rel wrel 4694    Fn wfn 5250  '''cafv 27972
This theorem is referenced by:  dfafn5b  28023  fnrnafv  28024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-dfat 27974  df-afv 27975
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