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Theorem dmxp 5226
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

Proof of Theorem dmxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 5010 . . 3
21dmeqi 5209 . 2
3 n0 3794 . . . . 5
43biimpi 194 . . . 4
54ralrimivw 2872 . . 3
6 dmopab3 5220 . . 3
75, 6sylib 196 . 2
82, 7syl5eq 2510 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  A.wral 2807   c0 3784  {copab 4509  X.cxp 5002  domcdm 5004
This theorem is referenced by:  dmxpid  5227  rnxp  5442  dmxpss  5443  ssxpb  5446  relrelss  5536  unixp  5545  xpexr2  6741  xpexcnv  6742  frxp  6910  mpt2curryd  7017  fodomr  7688  nqerf  9329  pwsbas  14884  pwsle  14889  imasaddfnlem  14925  imasvscafn  14934  efgrcl  16733  frlmip  18809  txindislem  20134  metustexhalfOLD  21066  metustexhalf  21067  rrxip  21822  dveq0  22401  dv11cn  22402  ismgmOLD  25322  mbfmcst  28230  eulerpartlemt  28310  0rrv  28390  bdayfo  29435  nobndlem3  29454  diophrw  30692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-dm 5014
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