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Theorem dfafn5a 28128
Description: Representation of a function in terms of its values, analogous to dffn5 5584 (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 5358 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 dfrel4v 5141 . . . 4  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
31, 2sylib 188 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
4 fnbr 5362 . . . . . . 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 2298 . . . . . . 7  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
8 fnbrafvb 28122 . . . . . . 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 4098 . . 3  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
133, 12eqtrd 2328 . 2  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) } )
14 df-mpt 4095 . 2  |-  ( x  e.  A  |->  ( F''' x ) )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F''' x ) ) }
1513, 14syl6eqr 2346 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   {copab 4092    e. cmpt 4093   Rel wrel 4710    Fn wfn 5266  '''cafv 28075
This theorem is referenced by:  dfafn5b  28129  fnrnafv  28130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-dfat 28077  df-afv 28078
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