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Theorem dffn5f 5774
Description: Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypothesis
Ref Expression
dffn5f.1  |-  F/_ x F
Assertion
Ref Expression
dffn5f  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem dffn5f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dffn5 5765 . 2  |-  ( F  Fn  A  <->  F  =  ( z  e.  A  |->  ( F `  z
) ) )
2 dffn5f.1 . . . . 5  |-  F/_ x F
3 nfcv 2572 . . . . 5  |-  F/_ x
z
42, 3nffv 5728 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2572 . . . 4  |-  F/_ z
( F `  x
)
6 fveq2 5721 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
74, 5, 6cbvmpt 4292 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  =  ( x  e.  A  |->  ( F `  x ) )
87eqeq2i 2446 . 2  |-  ( F  =  ( z  e.  A  |->  ( F `  z ) )  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
91, 8bitri 241 1  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   F/_wnfc 2559    e. cmpt 4259    Fn wfn 5442   ` cfv 5447
This theorem is referenced by:  prdsgsum  15545  fcomptf  24070  lgamgulm2  24813  refsum2cnlem1  27676
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 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fn 5450  df-fv 5455
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