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Theorem funbrafvb 27998
Description: Equivalence of function value and binary relation, analogous to funbrfvb 5771. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
funbrafvb  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F''' A )  =  B  <->  A F B ) )

Proof of Theorem funbrafvb
StepHypRef Expression
1 funfn 5484 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnbrafvb 27996 . 2  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  ( ( F''' A )  =  B  <-> 
A F B ) )
31, 2sylanb 460 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F''' A )  =  B  <->  A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214   dom cdm 4880   Fun wfun 5450    Fn wfn 5451  '''cafv 27950
This theorem is referenced by:  funbrafv  28000  funbrafv2b  28001  dfaimafn  28007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-dfat 27952  df-afv 27953
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