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Theorem fnbrafvb 28122
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 5579. (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 5359 . . . . . 6  |-  ( F  Fn  A  ->  dom  F  =  A )
2 eleq2 2357 . . . . . . . 8  |-  ( A  =  dom  F  -> 
( B  e.  A  <->  B  e.  dom  F ) )
32eqcoms 2299 . . . . . . 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 3775 . . . . . . 7  |-  ( B  e.  A  ->  { B }  C_  A )
87adantl 452 . . . . . 6  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { B }  C_  A )
9 fnssresb 5372 . . . . . . 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 5357 . . . . 5  |-  ( ( F  |`  { B } )  Fn  { B }  ->  Fun  ( F  |`  { B }
) )
1311, 12syl 15 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  ( F  |`  { B } ) )
14 df-dfat 28077 . . . . 5  |-  ( F defAt 
B  <->  ( B  e. 
dom  F  /\  Fun  ( F  |`  { B }
) ) )
15 afvfundmfveq 28106 . . . . 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 2304 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F''' B )  =  C  <->  ( F `  B )  =  C ) )
19 fnbrfvb 5579 . 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 1632    e. wcel 1696    C_ wss 3165   {csn 3653   class class class wbr 4039   dom cdm 4705    |` cres 4707   Fun wfun 5265    Fn wfn 5266   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem is referenced by:  fnopafvb  28123  funbrafvb  28124  dfafn5a  28128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-dfat 28077  df-afv 28078
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