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Theorem opelf 5737
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 5733 . . . 4  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
21sseld 3488 . . 3  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  <. C ,  D >.  e.  ( A  X.  B ) ) )
3 opelxp 5019 . . 3  |-  ( <. C ,  D >.  e.  ( A  X.  B
)  <->  ( C  e.  A  /\  D  e.  B ) )
42, 3syl6ib 226 . 2  |-  ( F : A --> B  -> 
( <. C ,  D >.  e.  F  ->  ( C  e.  A  /\  D  e.  B )
) )
54imp 429 1  |-  ( ( F : A --> B  /\  <. C ,  D >.  e.  F )  ->  ( C  e.  A  /\  D  e.  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1804   <.cop 4020    X. cxp 4987   -->wf 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-fun 5580  df-fn 5581  df-f 5582
This theorem is referenced by:  feu  5751  fcnvres  5752  fsn  6054
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