MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnbr Unicode version

Theorem fnbr 5489
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 5485 . . 3  |-  ( F  Fn  A  ->  Rel  F )
2 releldm 5044 . . 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 5486 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
54eleq2d 2456 . . 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 1717   class class class wbr 4155   dom cdm 4820   Rel wrel 4825    Fn wfn 5391
This theorem is referenced by:  fnop  5490  dffn5  5713  dffo4  5826  dffo5  5827  occllem  22655  chscllem2  22990  feqmptdf  23919  dfafn5a  27695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-dm 4830  df-fun 5398  df-fn 5399
  Copyright terms: Public domain W3C validator