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

Theorem fnop 5688
Description: The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
Assertion
Ref Expression
fnop  |-  ( ( F  Fn  A  /\  <. B ,  C >.  e.  F )  ->  B  e.  A )

Proof of Theorem fnop
StepHypRef Expression
1 df-br 4418 . 2  |-  ( B F C  <->  <. B ,  C >.  e.  F )
2 fnbr 5687 . 2  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
31, 2sylan2br 478 1  |-  ( ( F  Fn  A  /\  <. B ,  C >.  e.  F )  ->  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1867   <.cop 3999   class class class wbr 4417    Fn wfn 5587
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 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
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 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-dm 4855  df-fun 5594  df-fn 5595
This theorem is referenced by:  2elresin  5696  wfrlem12  7046  tfrlem9  7102  frrlem11  30354  poimirlem4  31692
  Copyright terms: Public domain W3C validator