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Theorem fnbr 5346
Description: The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
Assertion
Ref Expression
fnbr  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )

Proof of Theorem fnbr
StepHypRef Expression
1 fnrel 5342 . . 3  |-  ( F  Fn  A  ->  Rel  F )
2 releldm 4911 . . 3  |-  ( ( Rel  F  /\  B F C )  ->  B  e.  dom  F )
31, 2sylan 457 . 2  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  dom  F )
4 fndm 5343 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
54eleq2d 2350 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
65biimpa 470 . 2  |-  ( ( F  Fn  A  /\  B  e.  dom  F )  ->  B  e.  A
)
73, 6syldan 456 1  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   class class class wbr 4023   dom cdm 4689   Rel wrel 4694    Fn wfn 5250
This theorem is referenced by:  fnop  5347  dffn5  5568  dffo4  5676  dffo5  5677  occllem  21882  chscllem2  22217  feqmptdf  23228  dfafn5a  28022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-fun 5257  df-fn 5258
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