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Theorem dmxpss 5301
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 xpeq2 4894 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5292 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2485 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5073 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5084 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2485 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3657 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3399 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5089 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3401 . . 3  |-  ( dom  ( A  X.  B
)  =  A  ->  dom  ( A  X.  B
)  C_  A )
119, 10syl 16 . 2  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  C_  A )
128, 11pm2.61ine 2681 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    =/= wne 2600    C_ wss 3321   (/)c0 3629    X. cxp 4877   dom cdm 4879
This theorem is referenced by:  rnxpss  5302  ssxpb  5304  funssxp  5605  dff3  5883  fparlem3  6449  fparlem4  6450  brdom3  8407  brdom5  8408  brdom4  8409  canthwelem  8526  pwfseqlem4  8538  uzrdgfni  11299  rlimpm  12295  xpsc0  13786  xpsc1  13787  xpsfrnel2  13791  isohom  13998  ledm  14670  gsumxp  15551  dprd2d2  15603  tsmsxp  18185  dvbssntr  19788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889
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