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Theorem dmxpss 5675
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 5238 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 5662 . . . . . 6 (𝐴 × ∅) = ∅
31, 2syl6eq 2774 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5433 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5446 . . . 4 dom ∅ = ∅
64, 5syl6eq 2774 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4080 . . 3 ∅ ⊆ 𝐴
86, 7syl6eqss 3761 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5451 . . 3 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10 eqimss 3763 . . 3 (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
119, 10syl 17 . 2 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
128, 11pm2.61ine 2979 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1596  wne 2896  wss 3680  c0 4023   × cxp 5216  dom cdm 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228
This theorem is referenced by:  rnxpss  5676  ssxpb  5678  funssxp  6174  dff3  6487  fparlem3  7399  fparlem4  7400  brdom3  9463  brdom5  9464  brdom4  9465  canthwelem  9585  pwfseqlem4  9597  uzrdgfni  12872  xptrrel  13841  rlimpm  14351  xpsc0  16343  xpsc1  16344  xpsfrnel2  16348  isohom  16558  ledm  17346  gsumxp  18496  dprd2d2  18564  tsmsxp  22080  dvbssntr  23784  esum2d  30385  poimirlem3  33644  rtrclex  38343  trclexi  38346  rtrclexi  38347  cnvtrcl0  38352  dmtrcl  38353  rp-imass  38484  rfovcnvf1od  38717  issmflem  41359
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