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Theorem dmxp 4897
Description: The domain of a cross 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  X.  B )  =  A )

Proof of Theorem dmxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 4695 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
21dmeqi 4880 . 2  |-  dom  ( A  X.  B )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  B ) }
3 n0 3464 . . . . 5  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
43biimpi 186 . . . 4  |-  ( B  =/=  (/)  ->  E. x  x  e.  B )
54ralrimivw 2627 . . 3  |-  ( B  =/=  (/)  ->  A. y  e.  A  E. x  x  e.  B )
6 dmopab3 4891 . . 3  |-  ( A. y  e.  A  E. x  x  e.  B  <->  dom 
{ <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
75, 6sylib 188 . 2  |-  ( B  =/=  (/)  ->  dom  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
82, 7syl5eq 2327 1  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   (/)c0 3455   {copab 4076    X. cxp 4687   dom cdm 4689
This theorem is referenced by:  dmxpid  4898  rnxp  5106  dmxpss  5107  ssxpb  5110  xpexr2  5115  relrelss  5196  unixp  5205  frxp  6225  fodomr  7012  nqerf  8554  pwsbas  13386  pwsle  13391  imasaddfnlem  13430  imasvscafn  13439  efgrcl  15024  txindislem  17327  dveq0  19347  dv11cn  19348  ismgm  20987  mbfmcst  23564  0rrv  23654  bdayfo  24329  nobndlem3  24348  prjcp1  25084  cur1vald  25199  valcurfn1  25204  rngmgmbs3  25417  diophrw  26838  xpexcnv  27659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699
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