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Theorem funbrfvb 5430
Description: Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
Assertion
Ref Expression
funbrfvb  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )

Proof of Theorem funbrfvb
StepHypRef Expression
1 funfn 5121 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnbrfvb 5428 . 2  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  ( ( F `  A )  =  B  <->  A F B ) )
31, 2sylanb 280 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  =  B  <-> 
A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   class class class wbr 3897   dom cdm 4507   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  funbrfv2b  5432  dfimafn  5436  funimass4  5438  dvidlemap  12712
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