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Theorem fnbr 5539
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 5535 . . 3  |-  ( F  Fn  A  ->  Rel  F )
2 releldm 5094 . . 3  |-  ( ( Rel  F  /\  B F C )  ->  B  e.  dom  F )
31, 2sylan 458 . 2  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  dom  F )
4 fndm 5536 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
54eleq2d 2502 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
65biimpa 471 . 2  |-  ( ( F  Fn  A  /\  B  e.  dom  F )  ->  B  e.  A
)
73, 6syldan 457 1  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   class class class wbr 4204   dom cdm 4870   Rel wrel 4875    Fn wfn 5441
This theorem is referenced by:  fnop  5540  dffn5  5764  dffo4  5877  dffo5  5878  occllem  22797  chscllem2  23132  feqmptdf  24067  dfafn5a  27991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-dm 4880  df-fun 5448  df-fn 5449
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