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

Theorem dmxp 5252
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 5034 . . 3 (𝐴 × 𝐵) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
21dmeqi 5234 . 2 dom (𝐴 × 𝐵) = dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)}
3 n0 3889 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
43biimpi 204 . . . 4 (𝐵 ≠ ∅ → ∃𝑥 𝑥𝐵)
54ralrimivw 2949 . . 3 (𝐵 ≠ ∅ → ∀𝑦𝐴𝑥 𝑥𝐵)
6 dmopab3 5246 . . 3 (∀𝑦𝐴𝑥 𝑥𝐵 ↔ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
75, 6sylib 206 . 2 (𝐵 ≠ ∅ → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝑥𝐵)} = 𝐴)
82, 7syl5eq 2655 1 (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  wne 2779  wral 2895  c0 3873  {copab 4636   × cxp 5026  dom cdm 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-dm 5038
This theorem is referenced by:  dmxpid  5253  rnxp  5469  dmxpss  5470  ssxpb  5473  relrelss  5562  unixp  5571  xpexr2  6977  xpexcnv  6978  frxp  7151  mpt2curryd  7259  fodomr  7973  nqerf  9608  dmtrclfv  13553  pwsbas  15916  pwsle  15921  imasaddfnlem  15957  imasvscafn  15966  efgrcl  17897  frlmip  19878  txindislem  21188  metustexhalf  22112  rrxip  22903  dveq0  23484  dv11cn  23485  mbfmcst  29454  eulerpartlemt  29566  0rrv  29646  bdayfo  30880  nobndlem3  30899  curf  32353  curunc  32357  ismgmOLD  32615  diophrw  36136
  Copyright terms: Public domain W3C validator