MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmxpss Structured version   Visualization version   GIF version

Theorem dmxpss 6021
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 5569 . . . . . 6 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
2 xp0 6008 . . . . . 6 (𝐴 × ∅) = ∅
31, 2syl6eq 2869 . . . . 5 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
43dmeqd 5767 . . . 4 (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅)
5 dm0 5783 . . . 4 dom ∅ = ∅
64, 5syl6eq 2869 . . 3 (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅)
7 0ss 4347 . . 3 ∅ ⊆ 𝐴
86, 7eqsstrdi 4018 . 2 (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
9 dmxp 5792 . . 3 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
10 eqimss 4020 . . 3 (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴)
119, 10syl 17 . 2 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴)
128, 11pm2.61ine 3097 1 dom (𝐴 × 𝐵) ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wne 3013  wss 3933  c0 4288   × cxp 5546  dom cdm 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558
This theorem is referenced by:  rnxpss  6022  ssxpb  6024  funssxp  6528  dff3  6858  fparlem3  7798  fparlem4  7799  brdom3  9938  brdom5  9939  brdom4  9940  canthwelem  10060  pwfseqlem4  10072  uzrdgfni  13314  xptrrel  14328  rlimpm  14845  isohom  17034  ledm  17822  gsumxp  19025  dprd2d2  19095  tsmsxp  22690  dvbssntr  24425  esum2d  31251  poimirlem3  34776  rtrclex  39855  trclexi  39858  rtrclexi  39859  cnvtrcl0  39864  dmtrcl  39865  rp-imass  39995  rfovcnvf1od  40228  issmflem  42881
  Copyright terms: Public domain W3C validator