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Theorem dmxpss 6127
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 5658 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 6114 . . . . . 6 (𝐴 × ∅) = ∅
31, 2eqtrdi 2789 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5865 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5880 . . . 4 dom ∅ = ∅
64, 5eqtrdi 2789 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4360 . . 3 ∅ ⊆ 𝐴
86, 7eqsstrdi 4002 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5888 . . 3 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10 eqimss 4004 . . 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 1542  wne 2940  wss 3914  c0 4286   × cxp 5635  dom cdm 5637
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 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647
This theorem is referenced by:  rnxpss  6128  ssxpb  6130  resssxp  6226  funssxp  6701  dff3  7054  fparlem3  8050  fparlem4  8051  frxp2  8080  frxp3  8087  brdom3  10472  brdom5  10473  brdom4  10474  canthwelem  10594  pwfseqlem4  10606  uzrdgfni  13872  xptrrel  14874  rlimpm  15391  isohom  17667  ledm  18487  gsumxp  19761  dprd2d2  19831  tsmsxp  23529  dvbssntr  25287  gsumpart  31953  esum2d  32756  poimirlem3  36131  rtrclex  41981  trclexi  41984  rtrclexi  41985  cnvtrcl0  41990  dmtrcl  41991  rfovcnvf1od  42368  issmflem  45058  fvconstr  47012  fvconstrn0  47013  fvconstr2  47014  fvconst0ci  47015  fvconstdomi  47016
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