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Theorem dfafn5b 27687
Description: Representation of a function in terms of its values, analogous to dffn5 5704 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfafn5b  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F''' x ) ) ) )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    V( x)

Proof of Theorem dfafn5b
StepHypRef Expression
1 dfafn5a 27686 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
2 eqid 2380 . . . 4  |-  ( x  e.  A  |->  ( F''' x ) )  =  ( x  e.  A  |->  ( F''' x ) )
32fnmpt 5504 . . 3  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( x  e.  A  |->  ( F''' x ) )  Fn  A )
4 fneq1 5467 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F''' x ) )  ->  ( F  Fn  A  <->  ( x  e.  A  |->  ( F''' x ) )  Fn  A ) )
53, 4syl5ibrcom 214 . 2  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( F  =  ( x  e.  A  |->  ( F''' x ) )  ->  F  Fn  A )
)
61, 5impbid2 196 1  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F''' x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2642    e. cmpt 4200    Fn wfn 5382  '''cafv 27633
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-res 4823  df-iota 5351  df-fun 5389  df-fn 5390  df-fv 5395  df-dfat 27635  df-afv 27636
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