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Theorem dmxpss 5259
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 4852 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5250 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2452 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5031 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5042 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2452 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3616 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3358 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5047 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3360 . . 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 2643 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    =/= wne 2567    C_ wss 3280   (/)c0 3588    X. cxp 4835   dom cdm 4837
This theorem is referenced by:  rnxpss  5260  ssxpb  5262  funssxp  5563  dff3  5841  fparlem3  6407  fparlem4  6408  brdom3  8362  brdom5  8363  brdom4  8364  canthwelem  8481  pwfseqlem4  8493  uzrdgfni  11253  rlimpm  12249  xpsc0  13740  xpsc1  13741  xpsfrnel2  13745  isohom  13952  ledm  14624  gsumxp  15505  dprd2d2  15557  tsmsxp  18137  dvbssntr  19740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847
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