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Theorem dmxpss 6190
Description: The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.)
Assertion
Ref Expression
dmxpss dom (𝐴 × 𝐵) ⊆ 𝐴

Proof of Theorem dmxpss
StepHypRef Expression
1 xpeq2 5705 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 6177 . . . . . 6 (𝐴 × ∅) = ∅
31, 2eqtrdi 2792 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5915 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5930 . . . 4 dom ∅ = ∅
64, 5eqtrdi 2792 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4399 . . 3 ∅ ⊆ 𝐴
86, 7eqsstrdi 4027 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5938 . . 3 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10 eqimss 4041 . . 3 (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
119, 10syl 17 . 2 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
128, 11pm2.61ine 3024 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wne 2939  wss 3950  c0 4332   × cxp 5682  dom cdm 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694
This theorem is referenced by:  rnxpss  6191  ssxpb  6193  resssxp  6289  funssxp  6763  dff3  7119  fparlem3  8140  fparlem4  8141  frxp2  8170  frxp3  8177  brdom3  10569  brdom5  10570  brdom4  10571  canthwelem  10691  pwfseqlem4  10703  uzrdgfni  14000  xptrrel  15020  rlimpm  15537  isohom  17821  ledm  18636  gsumxp  19995  dprd2d2  20065  tsmsxp  24164  dvbssntr  25936  noseqrdgfn  28313  gsumpart  33061  esum2d  34095  poimirlem3  37631  rtrclex  43635  trclexi  43638  rtrclexi  43639  cnvtrcl0  43644  dmtrcl  43645  rfovcnvf1od  44022  issmflem  46747  fvconstr  48770  fvconstrn0  48771  fvconstr2  48772  fvconst0ci  48796  fvconstdomi  48797
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