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Theorem funfvbrb 5357
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 5356 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  <. A ,  ( F `
 A ) >.  e.  F )
2 df-br 3812 . . 3  |-  ( A F ( F `  A )  <->  <. A , 
( F `  A
) >.  e.  F )
31, 2sylibr 132 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  A F ( F `  A ) )
4 funrel 4986 . . 3  |-  ( Fun 
F  ->  Rel  F )
5 releldm 4628 . . 3  |-  ( ( Rel  F  /\  A F ( F `  A ) )  ->  A  e.  dom  F )
64, 5sylan 277 . 2  |-  ( ( Fun  F  /\  A F ( F `  A ) )  ->  A  e.  dom  F )
73, 6impbida 561 1  |-  ( Fun 
F  ->  ( A  e.  dom  F  <->  A F
( F `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   <.cop 3425   class class class wbr 3811   dom cdm 4401   Rel wrel 4406   Fun wfun 4963   ` cfv 4969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fn 4972  df-fv 4977
This theorem is referenced by:  fmptco  5406  climdm  10508
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