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Theorem dmxpss 4969
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 2689 . . . 4  |-  x  e. 
_V
21eldm2 4737 . . 3  |-  ( x  e.  dom  ( A  X.  B )  <->  E. y <. x ,  y >.  e.  ( A  X.  B
) )
3 opelxp1 4573 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  A )
43exlimiv 1577 . . 3  |-  ( E. y <. x ,  y
>.  e.  ( A  X.  B )  ->  x  e.  A )
52, 4sylbi 120 . 2  |-  ( x  e.  dom  ( A  X.  B )  ->  x  e.  A )
65ssriv 3101 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff set class
Syntax hints:   E.wex 1468    e. wcel 1480    C_ wss 3071   <.cop 3530    X. cxp 4537   dom cdm 4539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-dm 4549
This theorem is referenced by:  rnxpss  4970  dmxpss2  4971  ssxpbm  4974  ssxp1  4975  funssxp  5292  tfrlemibfn  6225  tfr1onlembfn  6241  tfrcllembfn  6254  frecuzrdgtcl  10199  frecuzrdgdomlem  10204  dvbssntrcntop  12838
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