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Theorem dmxpss 6171
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 5698 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 6158 . . . . . 6 (𝐴 × ∅) = ∅
31, 2eqtrdi 2789 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5906 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5921 . . . 4 dom ∅ = ∅
64, 5eqtrdi 2789 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4397 . . 3 ∅ ⊆ 𝐴
86, 7eqsstrdi 4037 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5929 . . 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 3026 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wne 2941  wss 3949  c0 4323   × cxp 5675  dom cdm 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687
This theorem is referenced by:  rnxpss  6172  ssxpb  6174  resssxp  6270  funssxp  6747  dff3  7102  fparlem3  8100  fparlem4  8101  frxp2  8130  frxp3  8137  brdom3  10523  brdom5  10524  brdom4  10525  canthwelem  10645  pwfseqlem4  10657  uzrdgfni  13923  xptrrel  14927  rlimpm  15444  isohom  17723  ledm  18543  gsumxp  19844  dprd2d2  19914  tsmsxp  23659  dvbssntr  25417  gsumpart  32207  esum2d  33091  poimirlem3  36491  rtrclex  42368  trclexi  42371  rtrclexi  42372  cnvtrcl0  42377  dmtrcl  42378  rfovcnvf1od  42755  issmflem  45443  fvconstr  47522  fvconstrn0  47523  fvconstr2  47524  fvconst0ci  47525  fvconstdomi  47526
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