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Theorem dfafn5b 27956
Description: Representation of a function in terms of its values, analogous to dffn5 5764 (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 27955 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
2 eqid 2435 . . . 4  |-  ( x  e.  A  |->  ( F''' x ) )  =  ( x  e.  A  |->  ( F''' x ) )
32fnmpt 5563 . . 3  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( x  e.  A  |->  ( F''' x ) )  Fn  A )
4 fneq1 5526 . . 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 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258    Fn wfn 5441  '''cafv 27903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-dfat 27905  df-afv 27906
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