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Theorem fnbrafvb 38088
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 5912. (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 5684 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
2 eleq2 2493 . . . . . . . 8  |-  ( A  =  dom  F  -> 
( B  e.  A  <->  B  e.  dom  F ) )
32eqcoms 2432 . . . . . . 7  |-  ( dom 
F  =  A  -> 
( B  e.  A  <->  B  e.  dom  F ) )
43biimpd 210 . . . . . 6  |-  ( dom 
F  =  A  -> 
( B  e.  A  ->  B  e.  dom  F
) )
51, 4syl 17 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  A  ->  B  e.  dom  F ) )
65imp 430 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B  e.  dom  F
)
7 snssi 4138 . . . . . . 7  |-  ( B  e.  A  ->  { B }  C_  A )
87adantl 467 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { B }  C_  A )
9 fnssresb 5697 . . . . . . 7  |-  ( F  Fn  A  ->  (
( F  |`  { B } )  Fn  { B }  <->  { B }  C_  A ) )
109adantr 466 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } )  Fn 
{ B }  <->  { B }  C_  A ) )
118, 10mpbird 235 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  Fn  { B } )
12 fnfun 5682 . . . . 5  |-  ( ( F  |`  { B } )  Fn  { B }  ->  Fun  ( F  |`  { B }
) )
1311, 12syl 17 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  ( F  |`  { B } ) )
14 df-dfat 38050 . . . . 5  |-  ( F defAt 
B  <->  ( B  e. 
dom  F  /\  Fun  ( F  |`  { B }
) ) )
15 afvfundmfveq 38072 . . . . 5  |-  ( F defAt 
B  ->  ( F''' B )  =  ( F `
 B ) )
1614, 15sylbir 216 . . . 4  |-  ( ( B  e.  dom  F  /\  Fun  ( F  |`  { B } ) )  ->  ( F''' B )  =  ( F `  B ) )
176, 13, 16syl2anc 665 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F''' B )  =  ( F `  B ) )
1817eqeq1d 2422 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  ( F `  B )  =  C ) )
19 fnbrfvb 5912 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2018, 19bitrd 256 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  B F C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    C_ wss 3433   {csn 3993   class class class wbr 4417   dom cdm 4845    |` cres 4847   Fun wfun 5586    Fn wfn 5587   ` cfv 5592   defAt wdfat 38047  '''cafv 38048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-dfat 38050  df-afv 38051
This theorem is referenced by:  fnopafvb  38089  funbrafvb  38090  dfafn5a  38094
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