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Theorem opelf 5062
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 5058 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
21sseld 2944 . . 3  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  <. C ,  D >.  e.  ( A  X.  B ) ) )
3 opelxp 4374 . . 3  |-  ( <. C ,  D >.  e.  ( A  X.  B
)  <->  ( C  e.  A  /\  D  e.  B ) )
42, 3syl6ib 150 . 2  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  ( C  e.  A  /\  D  e.  B )
) )
54imp 115 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 97    e. wcel 1393   <.cop 3378    X. cxp 4343   -->wf 4898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906
This theorem is referenced by:  feu  5072  fcnvres  5073  fsn  5335
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