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Theorem opelf 5442
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 5438 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
21sseld 3213 . . 3  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  <. C ,  D >.  e.  ( A  X.  B ) ) )
3 opelxp 4756 . . 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 1701   <.cop 3677    X. cxp 4724   -->wf 5288
This theorem is referenced by:  feu  5455  fcnvres  5456  fsn  5734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-dm 4736  df-rn 4737  df-fun 5294  df-fn 5295  df-f 5296
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