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Theorem fnbrafvb 28016
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 5563. (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 5343 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
2 eleq2 2344 . . . . . . . 8  |-  ( A  =  dom  F  -> 
( B  e.  A  <->  B  e.  dom  F ) )
32eqcoms 2286 . . . . . . 7  |-  ( dom 
F  =  A  -> 
( B  e.  A  <->  B  e.  dom  F ) )
43biimpd 198 . . . . . 6  |-  ( dom 
F  =  A  -> 
( B  e.  A  ->  B  e.  dom  F
) )
51, 4syl 15 . . . . 5  |-  ( F  Fn  A  ->  ( B  e.  A  ->  B  e.  dom  F ) )
65imp 418 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B  e.  dom  F
)
7 snssi 3759 . . . . . . 7  |-  ( B  e.  A  ->  { B }  C_  A )
87adantl 452 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { B }  C_  A )
9 fnssresb 5356 . . . . . . 7  |-  ( F  Fn  A  ->  (
( F  |`  { B } )  Fn  { B }  <->  { B }  C_  A ) )
109adantr 451 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } )  Fn 
{ B }  <->  { B }  C_  A ) )
118, 10mpbird 223 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  Fn  { B } )
12 fnfun 5341 . . . . 5  |-  ( ( F  |`  { B } )  Fn  { B }  ->  Fun  ( F  |`  { B }
) )
1311, 12syl 15 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  ( F  |`  { B } ) )
14 df-dfat 27974 . . . . 5  |-  ( F defAt 
B  <->  ( B  e. 
dom  F  /\  Fun  ( F  |`  { B }
) ) )
15 afvfundmfveq 28001 . . . . 5  |-  ( F defAt 
B  ->  ( F''' B )  =  ( F `
 B ) )
1614, 15sylbir 204 . . . 4  |-  ( ( B  e.  dom  F  /\  Fun  ( F  |`  { B } ) )  ->  ( F''' B )  =  ( F `  B ) )
176, 13, 16syl2anc 642 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F''' B )  =  ( F `  B ) )
1817eqeq1d 2291 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  ( F `  B )  =  C ) )
19 fnbrfvb 5563 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2018, 19bitrd 244 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  B F C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   class class class wbr 4023   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255   defAt wdfat 27971  '''cafv 27972
This theorem is referenced by:  fnopafvb  28017  funbrafvb  28018  dfafn5a  28022
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-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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