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Theorem dmxpss 5285
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 4866 . . . . . 6  |-  ( B  =  (/)  ->  ( A  X.  B )  =  ( A  X.  (/) ) )
2 xp0 5272 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
31, 2syl6eq 2480 . . . . 5  |-  ( B  =  (/)  ->  ( A  X.  B )  =  (/) )
43dmeqd 5054 . . . 4  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  dom  (/) )
5 dm0 5065 . . . 4  |-  dom  (/)  =  (/)
64, 5syl6eq 2480 . . 3  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  =  (/) )
7 0ss 3792 . . 3  |-  (/)  C_  A
86, 7syl6eqss 3515 . 2  |-  ( B  =  (/)  ->  dom  ( A  X.  B )  C_  A )
9 dmxp 5070 . . 3  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
10 eqimss 3517 . . 3  |-  ( dom  ( A  X.  B
)  =  A  ->  dom  ( A  X.  B
)  C_  A )
119, 10syl 17 . 2  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  C_  A )
128, 11pm2.61ine 2738 1  |-  dom  ( A  X.  B )  C_  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1438    =/= wne 2619    C_ wss 3437   (/)c0 3762    X. cxp 4849   dom cdm 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-xp 4857  df-rel 4858  df-cnv 4859  df-dm 4861
This theorem is referenced by:  rnxpss  5286  ssxpb  5288  funssxp  5757  dff3  6048  fparlem3  6907  fparlem4  6908  brdom3  8958  brdom5  8959  brdom4  8960  canthwelem  9077  pwfseqlem4  9089  uzrdgfni  12173  xptrrel  13038  rlimpm  13557  xpsc0  15459  xpsc1  15460  xpsfrnel2  15464  isohom  15674  ledm  16463  gsumxp  17601  dprd2d2  17670  tsmsxp  21161  dvbssntr  22847  esum2d  28916  poimirlem3  31901  rp-imass  36268
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