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Theorem dffn5f 5743
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 5734 . 2  |-  ( F  Fn  A  <->  F  =  ( z  e.  A  |->  ( F `  z
) ) )
2 dffn5f.1 . . . . 5  |-  F/_ x F
3 nfcv 2577 . . . . 5  |-  F/_ x
z
42, 3nffv 5695 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2577 . . . 4  |-  F/_ z
( F `  x
)
6 fveq2 5688 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
74, 5, 6cbvmpt 4379 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  =  ( x  e.  A  |->  ( F `  x ) )
87eqeq2i 2451 . 2  |-  ( F  =  ( z  e.  A  |->  ( F `  z ) )  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
91, 8bitri 249 1  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1364   F/_wnfc 2564    e. cmpt 4347    Fn wfn 5410   ` cfv 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423
This theorem is referenced by:  prdsgsum  16461  prdsgsumOLD  16462  fcomptf  25915  lgamgulm2  26952  refsum2cnlem1  29684
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