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Theorem opelf 5182
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 5178 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
21sseld 3024 . . 3  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  <. C ,  D >.  e.  ( A  X.  B ) ) )
3 opelxp 4467 . . 3  |-  ( <. C ,  D >.  e.  ( A  X.  B
)  <->  ( C  e.  A  /\  D  e.  B ) )
42, 3syl6ib 159 . 2  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  ( C  e.  A  /\  D  e.  B )
) )
54imp 122 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 102    e. wcel 1438   <.cop 3449    X. cxp 4436   -->wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019
This theorem is referenced by:  feu  5193  fcnvres  5194  fsn  5469
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