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Theorem dmxpss 5600
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 × 𝐵) ⊆ 𝐴

Proof of Theorem dmxpss
StepHypRef Expression
1 xpeq2 5163 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 5587 . . . . . 6 (𝐴 × ∅) = ∅
31, 2syl6eq 2701 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5358 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5371 . . . 4 dom ∅ = ∅
64, 5syl6eq 2701 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4005 . . 3 ∅ ⊆ 𝐴
86, 7syl6eqss 3688 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5376 . . 3 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10 eqimss 3690 . . 3 (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
119, 10syl 17 . 2 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
128, 11pm2.61ine 2906 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wne 2823  wss 3607  c0 3948   × cxp 5141  dom cdm 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153
This theorem is referenced by:  rnxpss  5601  ssxpb  5603  funssxp  6099  dff3  6412  fparlem3  7324  fparlem4  7325  brdom3  9388  brdom5  9389  brdom4  9390  canthwelem  9510  pwfseqlem4  9522  uzrdgfni  12797  xptrrel  13765  rlimpm  14275  xpsc0  16267  xpsc1  16268  xpsfrnel2  16272  isohom  16483  ledm  17271  gsumxp  18421  dprd2d2  18489  tsmsxp  22005  dvbssntr  23709  esum2d  30283  poimirlem3  33542  rtrclex  38241  trclexi  38244  rtrclexi  38245  cnvtrcl0  38250  dmtrcl  38251  rp-imass  38382  rfovcnvf1od  38615  issmflem  41257
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