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Theorem dmxpss 5266
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 4851 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5253 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2489 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5038 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5049 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2489 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3663 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3403 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5054 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3405 . . 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 2685 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    =/= wne 2604    C_ wss 3325   (/)c0 3634    X. cxp 4834   dom cdm 4836
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 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846
This theorem is referenced by:  rnxpss  5267  ssxpb  5269  funssxp  5568  dff3  5853  fparlem3  6673  fparlem4  6674  brdom3  8691  brdom5  8692  brdom4  8693  canthwelem  8813  pwfseqlem4  8825  uzrdgfni  11777  rlimpm  12974  xpsc0  14494  xpsc1  14495  xpsfrnel2  14499  isohom  14706  ledm  15390  gsumxp  16458  gsumxpOLD  16460  dprd2d2  16533  tsmsxp  19688  dvbssntr  21334
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