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Theorem fnbrafvb 27996
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 5769. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fnbrafvb  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  B F C ) )

Proof of Theorem fnbrafvb
StepHypRef Expression
1 fndm 5546 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
2 eleq2 2499 . . . . . . . 8  |-  ( A  =  dom  F  -> 
( B  e.  A  <->  B  e.  dom  F ) )
32eqcoms 2441 . . . . . . 7  |-  ( dom 
F  =  A  -> 
( B  e.  A  <->  B  e.  dom  F ) )
43biimpd 200 . . . . . 6  |-  ( dom 
F  =  A  -> 
( B  e.  A  ->  B  e.  dom  F
) )
51, 4syl 16 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  A  ->  B  e.  dom  F ) )
65imp 420 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B  e.  dom  F
)
7 snssi 3944 . . . . . . 7  |-  ( B  e.  A  ->  { B }  C_  A )
87adantl 454 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { B }  C_  A )
9 fnssresb 5559 . . . . . . 7  |-  ( F  Fn  A  ->  (
( F  |`  { B } )  Fn  { B }  <->  { B }  C_  A ) )
109adantr 453 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } )  Fn 
{ B }  <->  { B }  C_  A ) )
118, 10mpbird 225 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  Fn  { B } )
12 fnfun 5544 . . . . 5  |-  ( ( F  |`  { B } )  Fn  { B }  ->  Fun  ( F  |`  { B }
) )
1311, 12syl 16 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  ( F  |`  { B } ) )
14 df-dfat 27952 . . . . 5  |-  ( F defAt 
B  <->  ( B  e. 
dom  F  /\  Fun  ( F  |`  { B }
) ) )
15 afvfundmfveq 27980 . . . . 5  |-  ( F defAt 
B  ->  ( F''' B )  =  ( F `
 B ) )
1614, 15sylbir 206 . . . 4  |-  ( ( B  e.  dom  F  /\  Fun  ( F  |`  { B } ) )  ->  ( F''' B )  =  ( F `  B ) )
176, 13, 16syl2anc 644 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F''' B )  =  ( F `  B ) )
1817eqeq1d 2446 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  ( F `  B )  =  C ) )
19 fnbrfvb 5769 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2018, 19bitrd 246 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  B F C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   {csn 3816   class class class wbr 4214   dom cdm 4880    |` cres 4882   Fun wfun 5450    Fn wfn 5451   ` cfv 5456   defAt wdfat 27949  '''cafv 27950
This theorem is referenced by:  fnopafvb  27997  funbrafvb  27998  dfafn5a  28002
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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