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Theorem fnbr 5696
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 5692 . . 3  |-  ( F  Fn  A  ->  Rel  F )
2 releldm 5087 . . 3  |-  ( ( Rel  F  /\  B F C )  ->  B  e.  dom  F )
31, 2sylan 473 . 2  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  dom  F )
4 fndm 5693 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
54eleq2d 2499 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
65biimpa 486 . 2  |-  ( ( F  Fn  A  /\  B  e.  dom  F )  ->  B  e.  A
)
73, 6syldan 472 1  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1870   class class class wbr 4426   dom cdm 4854   Rel wrel 4859    Fn wfn 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-dm 4864  df-fun 5603  df-fn 5604
This theorem is referenced by:  fnop  5697  dffn5  5926  dffo4  6053  dffo5  6054  tfrlem5  7106  occllem  26791  chscllem2  27126  feqmptdf  28098  dfafn5a  38064
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