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Theorem dmxpss 6170
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 5697 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 6157 . . . . . 6 (𝐴 × ∅) = ∅
31, 2eqtrdi 2788 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5905 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5920 . . . 4 dom ∅ = ∅
64, 5eqtrdi 2788 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4396 . . 3 ∅ ⊆ 𝐴
86, 7eqsstrdi 4036 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5928 . . 3 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10 eqimss 4040 . . 3 (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
119, 10syl 17 . 2 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
128, 11pm2.61ine 3025 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wne 2940  wss 3948  c0 4322   × cxp 5674  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686
This theorem is referenced by:  rnxpss  6171  ssxpb  6173  resssxp  6269  funssxp  6746  dff3  7101  fparlem3  8099  fparlem4  8100  frxp2  8129  frxp3  8136  brdom3  10522  brdom5  10523  brdom4  10524  canthwelem  10644  pwfseqlem4  10656  uzrdgfni  13922  xptrrel  14926  rlimpm  15443  isohom  17722  ledm  18542  gsumxp  19843  dprd2d2  19913  tsmsxp  23658  dvbssntr  25416  gsumpart  32202  esum2d  33086  poimirlem3  36486  rtrclex  42358  trclexi  42361  rtrclexi  42362  cnvtrcl0  42367  dmtrcl  42368  rfovcnvf1od  42745  issmflem  45433  fvconstr  47512  fvconstrn0  47513  fvconstr2  47514  fvconst0ci  47515  fvconstdomi  47516
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