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Theorem dffn5imf 5586
Description: Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
Hypothesis
Ref Expression
dffn5imf.1  |-  F/_ x F
Assertion
Ref Expression
dffn5imf  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem dffn5imf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dffn5im 5576 . 2  |-  ( F  Fn  A  ->  F  =  ( z  e.  A  |->  ( F `  z ) ) )
2 dffn5imf.1 . . . 4  |-  F/_ x F
3 nfcv 2331 . . . 4  |-  F/_ x
z
42, 3nffv 5539 . . 3  |-  F/_ x
( F `  z
)
5 nfcv 2331 . . 3  |-  F/_ z
( F `  x
)
6 fveq2 5529 . . 3  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
74, 5, 6cbvmpt 4112 . 2  |-  ( z  e.  A  |->  ( F `
 z ) )  =  ( x  e.  A  |->  ( F `  x ) )
81, 7eqtrdi 2237 1  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363   F/_wnfc 2318    |-> cmpt 4078    Fn wfn 5225   ` cfv 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-iota 5192  df-fun 5232  df-fn 5233  df-fv 5238
This theorem is referenced by: (None)
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