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Theorem dffn5f 5713
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 5704 . 2  |-  ( F  Fn  A  <->  F  =  ( z  e.  A  |->  ( F `  z
) ) )
2 dffn5f.1 . . . . 5  |-  F/_ x F
3 nfcv 2516 . . . . 5  |-  F/_ x
z
42, 3nffv 5668 . . . 4  |-  F/_ x
( F `  z
)
5 nfcv 2516 . . . 4  |-  F/_ z
( F `  x
)
6 fveq2 5661 . . . 4  |-  ( z  =  x  ->  ( F `  z )  =  ( F `  x ) )
74, 5, 6cbvmpt 4233 . . 3  |-  ( z  e.  A  |->  ( F `
 z ) )  =  ( x  e.  A  |->  ( F `  x ) )
87eqeq2i 2390 . 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 1649   F/_wnfc 2503    e. cmpt 4200    Fn wfn 5382   ` cfv 5387
This theorem is referenced by:  prdsgsum  15472  fcomptf  23912  lgamgulm2  24592  refsum2cnlem1  27369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fn 5390  df-fv 5395
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