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Theorem dfafn5b 28023
Description: Representation of a function in terms of its values, analogous to dffn5 5568 (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 28022 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F''' x ) ) )
2 eqid 2283 . . . 4  |-  ( x  e.  A  |->  ( F''' x ) )  =  ( x  e.  A  |->  ( F''' x ) )
32fnmpt 5370 . . 3  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( x  e.  A  |->  ( F''' x ) )  Fn  A )
4 fneq1 5333 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F''' x ) )  ->  ( F  Fn  A  <->  ( x  e.  A  |->  ( F''' x ) )  Fn  A ) )
53, 4syl5ibrcom 213 . 2  |-  ( A. x  e.  A  ( F''' x )  e.  V  ->  ( F  =  ( x  e.  A  |->  ( F''' x ) )  ->  F  Fn  A )
)
61, 5impbid2 195 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 176    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077    Fn wfn 5250  '''cafv 27972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-dfat 27974  df-afv 27975
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