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Theorem dmxp 4913
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 4711 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
21dmeqi 4896 . 2  |-  dom  ( A  X.  B )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  B ) }
3 n0 3477 . . . . 5  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
43biimpi 186 . . . 4  |-  ( B  =/=  (/)  ->  E. x  x  e.  B )
54ralrimivw 2640 . . 3  |-  ( B  =/=  (/)  ->  A. y  e.  A  E. x  x  e.  B )
6 dmopab3 4907 . . 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 2340 1  |-  ( B  =/=  (/)  ->  dom  ( A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   {copab 4092    X. cxp 4703   dom cdm 4705
This theorem is referenced by:  dmxpid  4914  rnxp  5122  dmxpss  5123  ssxpb  5126  xpexr2  5131  relrelss  5212  unixp  5221  frxp  6241  fodomr  7028  nqerf  8570  pwsbas  13402  pwsle  13407  imasaddfnlem  13446  imasvscafn  13455  efgrcl  15040  txindislem  17343  dveq0  19363  dv11cn  19364  ismgm  21003  mbfmcst  23579  0rrv  23669  bdayfo  24400  nobndlem3  24419  prjcp1  25187  cur1vald  25302  valcurfn1  25307  rngmgmbs3  25520  diophrw  26941  xpexcnv  27762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715
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