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Theorem fnbr 5362
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 5358 . . 3  |-  ( F  Fn  A  ->  Rel  F )
2 releldm 4927 . . 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 5359 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
54eleq2d 2363 . . 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 1696   class class class wbr 4039   dom cdm 4705   Rel wrel 4710    Fn wfn 5266
This theorem is referenced by:  fnop  5363  dffn5  5584  dffo4  5692  dffo5  5693  occllem  21898  chscllem2  22233  feqmptdf  23243  dfafn5a  28128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-fun 5273  df-fn 5274
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