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Theorem dmxpss 5429
Description: The domain of a Cartesian 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 5007 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5416 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2517 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5196 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5207 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2517 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3807 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3547 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5212 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3549 . . 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 2773 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    =/= wne 2655    C_ wss 3469   (/)c0 3778    X. cxp 4990   dom cdm 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002
This theorem is referenced by:  rnxpss  5430  ssxpb  5432  funssxp  5735  dff3  6025  fparlem3  6875  fparlem4  6876  brdom3  8895  brdom5  8896  brdom4  8897  canthwelem  9017  pwfseqlem4  9029  uzrdgfni  12025  rlimpm  13272  xpsc0  14804  xpsc1  14805  xpsfrnel2  14809  isohom  15016  ledm  15700  gsumxp  16788  gsumxpOLD  16790  dprd2d2  16876  tsmsxp  20385  dvbssntr  22032  xptrrel  36660  rp-imass  36671
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