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Theorem fnbrafvb 31734
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 5908. (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 5680 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
2 eleq2 2540 . . . . . . . 8  |-  ( A  =  dom  F  -> 
( B  e.  A  <->  B  e.  dom  F ) )
32eqcoms 2479 . . . . . . 7  |-  ( dom 
F  =  A  -> 
( B  e.  A  <->  B  e.  dom  F ) )
43biimpd 207 . . . . . 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 429 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B  e.  dom  F
)
7 snssi 4171 . . . . . . 7  |-  ( B  e.  A  ->  { B }  C_  A )
87adantl 466 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { B }  C_  A )
9 fnssresb 5693 . . . . . . 7  |-  ( F  Fn  A  ->  (
( F  |`  { B } )  Fn  { B }  <->  { B }  C_  A ) )
109adantr 465 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } )  Fn 
{ B }  <->  { B }  C_  A ) )
118, 10mpbird 232 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  Fn  { B } )
12 fnfun 5678 . . . . 5  |-  ( ( F  |`  { B } )  Fn  { B }  ->  Fun  ( F  |`  { B }
) )
1311, 12syl 16 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  ( F  |`  { B } ) )
14 df-dfat 31696 . . . . 5  |-  ( F defAt 
B  <->  ( B  e. 
dom  F  /\  Fun  ( F  |`  { B }
) ) )
15 afvfundmfveq 31718 . . . . 5  |-  ( F defAt 
B  ->  ( F''' B )  =  ( F `
 B ) )
1614, 15sylbir 213 . . . 4  |-  ( ( B  e.  dom  F  /\  Fun  ( F  |`  { B } ) )  ->  ( F''' B )  =  ( F `  B ) )
176, 13, 16syl2anc 661 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F''' B )  =  ( F `  B ) )
1817eqeq1d 2469 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  ( F `  B )  =  C ) )
19 fnbrfvb 5908 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F `  B )  =  C  <-> 
B F C ) )
2018, 19bitrd 253 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  B F C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   {csn 4027   class class class wbr 4447   dom cdm 4999    |` cres 5001   Fun wfun 5582    Fn wfn 5583   ` cfv 5588   defAt wdfat 31693  '''cafv 31694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-dfat 31696  df-afv 31697
This theorem is referenced by:  fnopafvb  31735  funbrafvb  31736  dfafn5a  31740
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