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Theorem funfvbrb 5501
Description: Two ways to say that  A is in the domain of  F. (Contributed by Mario Carneiro, 1-May-2014.)
Assertion
Ref Expression
funfvbrb  |-  ( Fun 
F  ->  ( A  e.  dom  F  <->  A F
( F `  A
) ) )

Proof of Theorem funfvbrb
StepHypRef Expression
1 funfvop 5500 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 df-br 3900 . . 3  |-  ( A F ( F `  A )  <->  <. A , 
( F `  A
) >.  e.  F )
31, 2sylibr 133 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A F ( F `  A ) )
4 funrel 5110 . . 3  |-  ( Fun 
F  ->  Rel  F )
5 releldm 4744 . . 3  |-  ( ( Rel  F  /\  A F ( F `  A ) )  ->  A  e.  dom  F )
64, 5sylan 281 . 2  |-  ( ( Fun  F  /\  A F ( F `  A ) )  ->  A  e.  dom  F )
73, 6impbida 570 1  |-  ( Fun 
F  ->  ( A  e.  dom  F  <->  A F
( F `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   <.cop 3500   class class class wbr 3899   dom cdm 4509   Rel wrel 4514   Fun wfun 5087   ` cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fmptco  5554  climdm  11032  dvaddxx  12763  dvmulxx  12764  dviaddf  12765  dvimulf  12766  dvcjbr  12768
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