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Theorem dmxpss 5123
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
StepHypRef Expression
1 0ss 3496 . . 3  |-  (/)  C_  A
2 xpeq2 4720 . . . . . . 7  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
3 xp0 5114 . . . . . . 7  |-  ( A  X.  (/) )  =  (/)
42, 3syl6eq 2344 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
54dmeqd 4897 . . . . 5  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
6 dm0 4908 . . . . 5  |-  dom  (/)  =  (/)
75, 6syl6eq 2344 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
87sseq1d 3218 . . 3  |-  ( B  =  (/)  ->  ( dom  ( A  X.  B
)  C_  A  <->  (/)  C_  A
) )
91, 8mpbiri 224 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
10 dmxp 4913 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
11 eqimss 3243 . . 3  |-  ( dom  ( A  X.  B
)  =  A  ->  dom  ( A  X.  B
)  C_  A )
1210, 11syl 15 . 2  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  C_  A )
139, 12pm2.61ine 2535 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    =/= wne 2459    C_ wss 3165   (/)c0 3468    X. cxp 4703   dom cdm 4705
This theorem is referenced by:  rnxpss  5124  ssxpb  5126  funssxp  5418  dff3  5689  fparlem3  6236  fparlem4  6237  brdom3  8169  brdom5  8170  brdom4  8171  canthwelem  8288  pwfseqlem4  8300  uzrdgfni  11037  rlimpm  11990  xpsc0  13478  xpsc1  13479  xpsfrnel2  13483  isohom  13690  ledm  14362  gsumxp  15243  dprd2d2  15295  tsmsxp  17853  dvbssntr  19266  dmrngcmp  25854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715
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