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Theorem fnop 5481
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 4147 . 2  |-  ( B F C  <->  <. B ,  C >.  e.  F )
2 fnbr 5480 . 2  |-  ( ( F  Fn  A  /\  B F C )  ->  B  e.  A )
31, 2sylan2br 463 1  |-  ( ( F  Fn  A  /\  <. B ,  C >.  e.  F )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   <.cop 3753   class class class wbr 4146    Fn wfn 5382
This theorem is referenced by:  2elresin  5489  tfrlem2  6566  tfrlem9  6575  wfrlem12  25284  frrlem11  25310
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-dm 4821  df-fun 5389  df-fn 5390
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