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Theorem dmxpss 4861
Description: The domain of a cross product is a subclass of the first factor. (Contributed by NM, 19-Mar-2007.)
Assertion
Ref Expression
dmxpss  |-  dom  ( A  X.  B )  C_  A

Proof of Theorem dmxpss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2622 . . . 4  |-  x  e. 
_V
21eldm2 4634 . . 3  |-  ( x  e.  dom  ( A  X.  B )  <->  E. y <. x ,  y >.  e.  ( A  X.  B
) )
3 opelxp1 4471 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  A )
43exlimiv 1534 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  X.  B )  ->  x  e.  A )
52, 4sylbi 119 . 2  |-  ( x  e.  dom  ( A  X.  B )  ->  x  e.  A )
65ssriv 3029 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff set class
Syntax hints:   E.wex 1426    e. wcel 1438    C_ wss 2999   <.cop 3449    X. cxp 4436   dom cdm 4438
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-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-dm 4448
This theorem is referenced by:  rnxpss  4862  dmxpss2  4863  ssxpbm  4866  ssxp1  4867  funssxp  5180  tfrlemibfn  6093  tfr1onlembfn  6109  tfrcllembfn  6122  frecuzrdgtcl  9819  frecuzrdgdomlem  9824
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