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

Theorem dmxp 5928
Description: The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmxp (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)

Proof of Theorem dmxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 5682 . . 3 (𝐴 × 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
21dmeqi 5904 . 2 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
3 n0 4346 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
43biimpi 215 . . . 4 (𝐵 ≠ ∅ → ∃𝑥 𝑥𝐵)
54ralrimivw 3149 . . 3 (𝐵 ≠ ∅ → ∀𝑦𝐴𝑥 𝑥𝐵)
6 dmopab3 5919 . . 3 (∀𝑦𝐴𝑥 𝑥𝐵 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
75, 6sylib 217 . 2 (𝐵 ≠ ∅ → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
82, 7eqtrid 2783 1 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1780  wcel 2105  wne 2939  wral 3060  c0 4322  {copab 5210   × cxp 5674  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686
This theorem is referenced by:  dmxpid  5929  rnxp  6169  dmxpss  6170  ssxpb  6173  relrelss  6272  unixp  6281  xpexr2  7914  xpexcnv  7915  frxp  8117  mpocurryd  8260  fodomr  9134  nqerf  10931  dmtrclfv  14972  pwsbas  17440  pwsle  17445  imasaddfnlem  17481  imasvscafn  17490  efgrcl  19631  frlmip  21643  txindislem  23457  metustexhalf  24385  rrxip  25238  dveq0  25853  dv11cn  25854  noxp1o  27509  noextendseq  27513  bdayfo  27523  noetasuplem2  27580  noetasuplem4  27582  noetainflem2  27584  noetainflem4  27586  mbfmcst  33722  eulerpartlemt  33834  0rrv  33914  curf  36930  curunc  36934  ismgmOLD  37182  diophrw  41960  onnog  42643  onnobdayg  42644  bdaybndbday  42646
  Copyright terms: Public domain W3C validator