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Theorem dmxp 4850
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
StepHypRef Expression
1 df-xp 4640 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
21dmeqi 4833 . 2  |-  dom  (  A  X.  B )  =  dom  { <. y ,  x >.  |  (
y  e.  A  /\  x  e.  B ) }
3 n0 3406 . . . . 5  |-  ( B  =/=  (/)  <->  E. x  x  e.  B )
43biimpi 188 . . . 4  |-  ( B  =/=  (/)  ->  E. x  x  e.  B )
54ralrimivw 2598 . . 3  |-  ( B  =/=  (/)  ->  A. y  e.  A  E. x  x  e.  B )
6 dmopab3 4844 . . 3  |-  ( A. y  e.  A  E. x  x  e.  B  <->  dom 
{ <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
75, 6sylib 190 . 2  |-  ( B  =/=  (/)  ->  dom  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }  =  A )
82, 7syl5eq 2300 1  |-  ( B  =/=  (/)  ->  dom  (  A  X.  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   (/)c0 3397   {copab 4016    X. cxp 4624   dom cdm 4626
This theorem is referenced by:  dmxpid  4851  rnxp  5059  dmxpss  5060  ssxpb  5063  xpexr2  5068  relrelss  5148  unixp  5157  frxp  6124  fodomr  6945  nqerf  8487  pwsbas  13313  pwsle  13318  imasaddfnlem  13357  imasvscafn  13366  efgrcl  14951  txindislem  17254  dveq0  19274  dv11cn  19275  ismgm  20912  axbday  23662  axfelem3  23682  prjcp1  24415  cur1vald  24531  valcurfn1  24536  rngmgmbs3  24749  diophrw  26170  xpexcnv  26992
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-dm 4644
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