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Theorem opelf 5404
Description: The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelf  |-  ( ( F : A --> B  /\  <. C ,  D >.  e.  F )  ->  ( C  e.  A  /\  D  e.  B )
)

Proof of Theorem opelf
StepHypRef Expression
1 fssxp 5400 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
21sseld 3179 . . 3  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  <. C ,  D >.  e.  ( A  X.  B ) ) )
3 opelxp 4719 . . 3  |-  ( <. C ,  D >.  e.  ( A  X.  B
)  <->  ( C  e.  A  /\  D  e.  B ) )
42, 3syl6ib 217 . 2  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  ( C  e.  A  /\  D  e.  B )
) )
54imp 418 1  |-  ( ( F : A --> B  /\  <. C ,  D >.  e.  F )  ->  ( C  e.  A  /\  D  e.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   <.cop 3643    X. cxp 4687   -->wf 5251
This theorem is referenced by:  feu  5417  fcnvres  5418  fsn  5696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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