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Theorem dffn5f 5850
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 5841 . 2  |-  ( F  Fn  A  <->  F  =  ( z  e.  A  |->  ( F `  z
) ) )
2 dffn5f.1 . . . . 5  |-  F/_ x F
3 nfcv 2614 . . . . 5  |-  F/_ x
z
42, 3nffv 5801 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2614 . . . 4  |-  F/_ z
( F `  x
)
6 fveq2 5794 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
74, 5, 6cbvmpt 4485 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  =  ( x  e.  A  |->  ( F `  x ) )
87eqeq2i 2470 . 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 1370   F/_wnfc 2600    |-> cmpt 4453    Fn wfn 5516   ` cfv 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529
This theorem is referenced by:  prdsgsum  16588  prdsgsumOLD  16589  fcomptf  26126  lgamgulm2  27161  refsum2cnlem1  29902
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