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Theorem dfafn5a 27695
Description: Representation of a function in terms of its values, analogous to dffn5 5713 (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 5485 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5264 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 189 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5489 . . . . . . 7  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 424 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 617 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2391 . . . . . . 7  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
8 fnbrafvb 27689 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F''' x )  =  y  <->  x F
y ) )
97, 8syl5bb 249 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F''' x )  <->  x F
y ) )
109pm5.32da 623 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F''' x ) )  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 248 . . . 4  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F''' x ) ) ) )
1211opabbidv 4214 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
133, 12eqtrd 2421 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
14 df-mpt 4211 . 2  |-  ( x  e.  A  |->  ( F''' x ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) }
1513, 14syl6eqr 2439 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4155   {copab 4208    e. cmpt 4209   Rel wrel 4825    Fn wfn 5391  '''cafv 27642
This theorem is referenced by:  dfafn5b  27696  fnrnafv  27697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-dfat 27644  df-afv 27645
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