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Theorem dfafn5a 32406
Description: Representation of a function in terms of its values, analogous to dffn5 5918 (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 5685 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5464 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 196 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5689 . . . . . . 7  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
54ex 434 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
65pm4.71rd 635 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
7 eqcom 2466 . . . . . . 7  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
8 fnbrafvb 32400 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F''' x )  =  y  <->  x F
y ) )
97, 8syl5bb 257 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F''' x )  <->  x F
y ) )
109pm5.32da 641 . . . . 5  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F''' x ) )  <->  ( x  e.  A  /\  x F y ) ) )
116, 10bitr4d 256 . . . 4  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F''' x ) ) ) )
1211opabbidv 4520 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
133, 12eqtrd 2498 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
14 df-mpt 4517 . 2  |-  ( x  e.  A  |->  ( F''' x ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) }
1513, 14syl6eqr 2516 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456   {copab 4514    |-> cmpt 4515   Rel wrel 5013    Fn wfn 5589  '''cafv 32360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-dfat 32362  df-afv 32363
This theorem is referenced by:  dfafn5b  32407  fnrnafv  32408
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