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Theorem dfafn5a 27991
Description: Representation of a function in terms of its values, analogous to dffn5 5764 (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 5535 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5314 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 189 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5539 . . . . . . 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 2437 . . . . . . 7  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
8 fnbrafvb 27985 . . . . . . 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 4263 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
133, 12eqtrd 2467 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
14 df-mpt 4260 . 2  |-  ( x  e.  A  |->  ( F''' x ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) }
1513, 14syl6eqr 2485 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   {copab 4257    e. cmpt 4258   Rel wrel 4875    Fn wfn 5441  '''cafv 27939
This theorem is referenced by:  dfafn5b  27992  fnrnafv  27993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-dfat 27941  df-afv 27942
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